Real Forms and Real Semisimple Lie Algebras — Examples

Given: ${\mathrm{A}\mathrm{C}}_{\omega }$ and $\mathfrak{s}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with the basis $e=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),f=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),h=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),$ so that $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$, and the two maps ${\sigma }_c(X)=-X^{\ast }$ and ${\sigma }_s(X)=\bar{X}$.

${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice; it is used only through the matrix Lie-group examples cited in [L1] and [L2].

$\mathfrak{s}$ is the Lie algebra of traceless complex $2\times 2$ matrices with bracket $[A,B]=AB-BA$, with basis $e,f,h$ and the displayed relations, and the real traceless matrices form the real Lie subalgebra ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ (The special linear Lie algebra sl_2, General and special linear Lie groups).

$\mathfrak{s}\mathfrak{u}(2)=\{X\in M_2(\mathbb{C}):X^{\ast }+X=0,\mathrm{tr}X=0\}$ is a real Lie subalgebra of $\mathfrak{s}$ with the same bracket (Unitary and special unitary Lie groups).

If $\sigma$ is a conjugate-linear involutive automorphism of a finite-dimensional complex Lie algebra $\mathfrak{g}$, then its fixed locus ${\mathfrak{g}}^{\sigma }$ is a real form of $\mathfrak{g}$ (Real forms correspond to conjugate-linear involutions, Real form of a complex Lie algebra).

The Killing form of ${\mathfrak{s}\mathfrak{l}}_2$ satisfies $B(h,h)=8$, $B(e,f)=B(f,e)=4$, and all other pairings of basis vectors zero; equivalently $B(X,Y)=4\mathrm{tr}(XY)$ for all $X,Y$ (Killing form of sl_2, Killing form).

When the ambient complex Lie algebra is finite-dimensional and semisimple, a real form ${\mathfrak{g}}_0$ is a compact real form exactly when $B(X,X)<0$ for every nonzero $X\in {\mathfrak{g}}_0$; it is a split real form exactly when it contains a Cartan subalgebra ${\mathfrak{h}}_0$ such that every ${\mathrm{ad}}_H$, $H\in {\mathfrak{h}}_0$, is diagonalizable over $\mathbb{R}$ (Compact real form of a complex semisimple Lie algebra, Split real form, Cartan subalgebra).

The conjugate transpose $X^{\ast }={\bar{X}}^{\mathsf{T}}$ is additive, and $(XY{)}^{\ast }=Y^{\ast }{X}^{\ast }$ (The transpose $A^{\mathsf{T}}$ of a matrix).

Proof technique: direct matrix computation.

1.1 Both maps ${\sigma }_c$ and ${\sigma }_s$ are real-linear, additive, involutive and conjugate-linear, and they preserve brackets. By [L6] the conjugate transpose reverses products, ${\sigma }_c(AB)=-(AB{)}^{\ast }=-B^{\ast }{A}^{\ast }$, while ${\sigma }_c(A){\sigma }_c(B)=A^{\ast }{B}^{\ast }$; taking the difference gives ${\sigma }_c[A,B]=-(AB-BA{)}^{\ast }=A^{\ast }{B}^{\ast }-B^{\ast }{A}^{\ast }=[{\sigma }_{c}A,{\sigma }_{c}B]$. Entrywise conjugation is multiplicative, $\overline{AB}=\bar{A}\bar{B}$, so ${\sigma }_s[A,B]=[{\sigma }_{s}A,{\sigma }_{s}B]$. [given, L6, algebra]

1.2 The fixed algebra of ${\sigma }_c$ inside $\mathfrak{s}$ is $\mathfrak{s}\mathfrak{u}(2)$: a traceless $X$ satisfies $X=-X^{\ast }$ exactly when $X$ is skew-Hermitian, so ${\mathfrak{s}}^{{\sigma }_c}=\{X\in M_2(\mathbb{C}):X^{\ast }+X=0,\mathrm{tr}X=0\}$, which is $\mathfrak{s}\mathfrak{u}(2)$ by [L2]. [given, L2, algebra]

1.3 The fixed algebra of ${\sigma }_s$ is ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$: a matrix is fixed by entrywise conjugation exactly when it is real, and a real traceless matrix lies in ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ by [L1]. [given, L1, algebra]

1.4 The Killing form of $\mathfrak{s}$ is $B(X,Y)=4\mathrm{tr}(XY)$. Both sides are symmetric bilinear, so it suffices to compare them on basis pairs: $4\mathrm{tr}(ef)=4=B(e,f)$, $4\mathrm{tr}(h^2)=8=B(h,h)$, and $4\mathrm{tr}(e^2)=4\mathrm{tr}(f^2)=4\mathrm{tr}(eh)=4\mathrm{tr}(fh)=0$, matching the vanishing pairings of [L4]. In the basis $(e,f,h)$ its Gram matrix is $\left(\begin{array}{ccc}0& 4& 0\\ 4& 0& 0\\ 0& 0& 8\end{array}\right)$, whose determinant is $-128\mathrm{\ne }0$; hence $B$ is nondegenerate and $\mathfrak{s}$ is semisimple by Cartan's criterion over the characteristic-zero field $\mathbb{C}$. Thus the ambient hypothesis in [L5] has been established before either definition is invoked. [L4, Cartan's semisimplicity criterion, algebra]

1.5 The line $\mathbb{R}h$ is a Cartan subalgebra of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$: it is abelian, hence nilpotent, and its normalizer is itself because $[h,X]=0$ for $X=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$ forces $2b=0$ and $-2c=0$, so $X\in \mathbb{R}h$. Since $[h,h]=0$, $[h,e]=2e$ and $[h,f]=-2f$ by the given relations, ${\mathrm{ad}}_h$ is diagonal on the basis $(h,e,f)$ with real eigenvalues $0,2,-2$, so ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ is a split real form by [L5]. [given, L1, L5, algebra]

2.1 Both fixed loci are real forms. Every $Z\in \mathfrak{s}$ decomposes as $Z=X+iY$ with $X=\frac{1}{2}(Z+\bar{Z})$ and $Y=\frac{1}{2i}(Z-\bar{Z})$ real traceless, so ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ spans $\mathfrak{s}$ over $\mathbb{C}$ and has real dimension $3={\dim }_{\mathbb{C}}\mathfrak{s}$; alternatively this is the general conclusion of [L3] applied to the involution ${\sigma }_s$ of step 1.1. Likewise every $Z\in \mathfrak{s}$ is $Z=X+iY$ with $X=\frac{1}{2}(Z-Z^{\ast })$ and $Y=\frac{1}{2i}(Z+Z^{\ast })$ both skew-Hermitian and traceless, so $\mathfrak{s}\mathfrak{u}(2)$ spans $\mathfrak{s}$ over $\mathbb{C}$ and has real dimension $3={\dim }_{\mathbb{C}}\mathfrak{s}$; again [L3] gives the same conclusion from ${\sigma }_c$. [L3, step 1.1, step 1.2, step 1.3, algebra]

2.2 For nonzero $X\in \mathfrak{s}\mathfrak{u}(2)$ one has $X^{\ast }=-X$, so step 1.4 gives $B(X,X)=4\mathrm{tr}(X^2)=-4\mathrm{tr}(X{X}^{\ast })=-4{\sum }_{i,j}\mathrm{\mid }{X}_{ij}{\mathrm{\mid }}^2<0$, because a nonzero matrix has a nonzero entry. Hence $B$ is negative definite on $\mathfrak{s}\mathfrak{u}(2)$, and $\mathfrak{s}\mathfrak{u}(2)$ is a compact real form by [L5]. [step 1.4, L5, algebra]

3.1 The two forms are genuinely different: $B(h,h)=8>0$ on ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ while $B(e-f,e-f)=B(e,e)-2B(e,f)+B(f,f)=-8<0$, so the Killing form of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ is indefinite, as a noncompact real form must be, whereas the form on $\mathfrak{s}\mathfrak{u}(2)$ is definite by step 2.2. All computations are finite; ${\mathrm{A}\mathrm{C}}_{\omega }$ enters only through [L1] and [L2]. [A1, step 1.4, step 2.2, step 1.5, algebra] ∎

Given: An integer $n\ge 2$, the real Lie algebra ${\mathfrak{g}}_0={\mathfrak{s}\mathfrak{l}}_n(\mathbb{R})$ of real traceless matrices, the map $\theta (X)=-X^{\mathsf{T}}$, and the Killing form $B$ of ${\mathfrak{g}}_0$.

${\mathfrak{s}\mathfrak{l}}_n(\mathbb{R})$ is a real Lie subalgebra of $M_n(\mathbb{R})$ under $[X,Y]=XY-YX$, with $\mathrm{tr}(XY)=\mathrm{tr}(YX)$ and $\mathrm{tr}X=0$ for every $X$ (General and special linear Lie groups).

Transposition is additive, involutive and reverses products: $(XY{)}^{\mathsf{T}}=Y^{\mathsf{T}}{X}^{\mathsf{T}}$ (The transpose $A^{\mathsf{T}}$ of a matrix).

The Killing form of ${\mathfrak{s}\mathfrak{l}}_n$ is $B(X,Y)=2n\mathrm{tr}(XY)$ for $n\ge 2$, so it is nondegenerate on ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{R})$, and ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{R})$ is therefore semisimple (Classical simple Lie algebras and their Killing forms, Killing form, Cartan's semisimplicity criterion).

The orthogonal Lie algebra is $\mathfrak{s}\mathfrak{o}(n)=\{X\in M_n(\mathbb{R}):X^{\mathsf{T}}+X=0\}$ (Orthogonal and special orthogonal Lie groups).

A Cartan involution of a real semisimple Lie algebra ${\mathfrak{g}}_0$ is an involutive automorphism $\theta$ with $B_{\theta }(X,Y)=-B(X,\theta Y)$ positive definite, and its Cartan decomposition is the decomposition into the $+1$ and $-1$ eigenspaces; then $[{\mathfrak{k}}_0,{\mathfrak{k}}_0]\subseteq {\mathfrak{k}}_0$, $[{\mathfrak{k}}_0,{\mathfrak{p}}_0]\subseteq {\mathfrak{p}}_0$, $[{\mathfrak{p}}_0,{\mathfrak{p}}_0]\subseteq {\mathfrak{k}}_0$, with $B$ negative definite on ${\mathfrak{k}}_0$ and positive definite on ${\mathfrak{p}}_0$ (Cartan involution of a real semisimple Lie algebra, Cartan decomposition of a real semisimple Lie algebra, Bracket relations and Killing signs in a Cartan decomposition).

Proof technique: direct matrix computation.

1.1 The map $\theta$ is an involutive automorphism of ${\mathfrak{g}}_0$: by [L2], ${\theta }^{2}X=X^{\mathsf{T}\mathsf{T}}=X$; $\mathrm{tr}(-X^{\mathsf{T}})=-\mathrm{tr}X$ makes $\theta$ preserve tracelessness; and $\theta [X,Y]=-(XY-YX{)}^{\mathsf{T}}=-(Y^{\mathsf{T}}{X}^{\mathsf{T}}-X^{\mathsf{T}}{Y}^{\mathsf{T}})=[\theta X,\theta Y]$ by [L2]. [given, L1, L2, algebra]

1.2 The Killing form satisfies $B(X,Y)=2n\mathrm{tr}(XY)$ on all of ${\mathfrak{g}}_0$ by [L3]. [L3]

2.1 The fixed space of $\theta$ is ${\mathfrak{k}}_0=\mathfrak{s}\mathfrak{o}(n)$: $\theta X=X$ means $-X^{\mathsf{T}}=X$, that is $X^{\mathsf{T}}+X=0$, which is the defining condition of $\mathfrak{s}\mathfrak{o}(n)$ by [L4]; such an $X$ automatically has $\mathrm{tr}X=0$, so no tracelessness is lost. [step 1.1, L1, L4, algebra]

2.2 The anti-fixed space of $\theta$ is the space of symmetric traceless matrices: $\theta X=-X$ means $-X^{\mathsf{T}}=-X$, that is $X^{\mathsf{T}}=X$, and membership in ${\mathfrak{g}}_0$ adds $\mathrm{tr}X=0$. [step 1.1, L1, algebra]

2.3 The form $B_{\theta }(X,Y)=-B(X,\theta Y)$ is positive definite: by steps 1.2 and 2.2, $B_{\theta }(X,Y)=-2n\mathrm{tr}(-X{Y}^{\mathsf{T}})=2n\mathrm{tr}(X{Y}^{\mathsf{T}})=2n{\sum }_{i,j=1}^n{X}_{ij}{Y}_{ij}$, and $\mathrm{tr}(X{X}^{\mathsf{T}})={\sum }_{i,j}{X}_{ij}^2>0$ for $X\mathrm{\ne }0$. Hence $\theta$ is a Cartan involution of the semisimple algebra ${\mathfrak{g}}_0$ of [L3]. [step 1.1, step 1.2, L3, L5, algebra]

3.1 The eigenspace decomposition ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ holds with ${\mathfrak{k}}_0$ as in step 2.1 and ${\mathfrak{p}}_0$ as in step 2.2, since every $X$ is $\frac{1}{2}(X-\theta X)+\frac{1}{2}(X+\theta X)$ and the two summands are respectively symmetric and skew-symmetric. [step 2.1, step 2.2, algebra]

3.2 The bracket relations follow directly from transposition: for skew $X,Y$ one has $[X,Y{]}^{\mathsf{T}}=-[X,Y]$, for skew $X$ and symmetric $Y$ one has $[X,Y{]}^{\mathsf{T}}=[X,Y]$, and for symmetric $X,Y$ one has $[X,Y{]}^{\mathsf{T}}=-[X,Y]$; hence $[{\mathfrak{k}}_0,{\mathfrak{k}}_0]\subseteq {\mathfrak{k}}_0$, $[{\mathfrak{k}}_0,{\mathfrak{p}}_0]\subseteq {\mathfrak{p}}_0$ and $[{\mathfrak{p}}_0,{\mathfrak{p}}_0]\subseteq {\mathfrak{k}}_0$, in agreement with [L5]. [step 2.1, step 2.2, L5, algebra]

3.3 The Killing signs also follow from the computations: for $X\in {\mathfrak{k}}_0$ we have $B(X,X)=2n\mathrm{tr}(X^2)=-2n\mathrm{tr}(X{X}^{\mathsf{T}})<0$ for $X\mathrm{\ne }0$, and for $Y\in {\mathfrak{p}}_0$ we have $B(Y,Y)=2n\mathrm{tr}(Y^2)=2n\mathrm{tr}(Y{Y}^{\mathsf{T}})>0$ for $Y\mathrm{\ne }0$, so $B$ is negative definite on ${\mathfrak{k}}_0$ and positive definite on ${\mathfrak{p}}_0$. [step 1.2, step 2.1, step 2.2, algebra]

4.1 Endpoints and scope: for $n=1$ the algebra ${\mathfrak{s}\mathfrak{l}}_1(\mathbb{R})=0$ is semisimple (its Killing form is nondegenerate vacuously), and the same construction degenerates to the zero Cartan decomposition. The hypothesis $n\ge 2$ isolates the nonzero classical case covered by [L3]. For $n=2$ one has $\dim {\mathfrak{k}}_0=1$ and $\dim {\mathfrak{p}}_0=2$, so both summands are nonzero. The computations are finite, use no choice principle, and the displayed identification of ${\mathfrak{k}}_0$ with $\mathfrak{s}\mathfrak{o}(n)$ is an equality of matrix sets, not merely an isomorphism. [given, step 2.1, step 2.2, L3, algebra] ∎

Given: The Axiom of Choice; an integer $n\ge 2$; the Lie group ${\mathrm{SL}}_n(\mathbb{R})$ with Lie algebra ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{R})$; the Cartan decomposition ${\mathfrak{g}}_0={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ with ${\mathfrak{k}}_0=\mathfrak{s}\mathfrak{o}(n)$ and ${\mathfrak{p}}_0$ the symmetric traceless matrices of Cartan involution and k plus p for sl n r; and an element $g\in {\mathrm{SL}}_n(\mathbb{R})$.

The Axiom of Choice is The Axiom of Choice; it enters only through the global Cartan decomposition of [L5] and the smooth structure of ${\mathrm{SL}}_n(\mathbb{R})$.

${\mathrm{SL}}_n(\mathbb{R})=\{A:\det A=1\}$ is an embedded Lie subgroup of ${\mathrm{GL}}_n(\mathbb{R})$ with Lie algebra ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{R})$, and $\mathrm{SO}(n)$ is the determinant-one subgroup of $O(n)$ with Lie algebra $\mathfrak{s}\mathfrak{o}(n)$ (General and special linear Lie groups, Orthogonal and special orthogonal Lie groups).

Every endomorphism $T$ of a finite-dimensional real inner product space has a polar decomposition $T=SU$ with $U=\sqrt{T^{\ast }T}$ non-negative and $S$ an isometry on the orthogonal complement of $\ker T$; the factor $U$ is unique and $S$ is unique when $T$ is invertible (Every endomorphism has a polar decomposition T = SU with U non-negative and S an isometry on the orthogonal complement of ker T, and S is unique exactly when T is invertible).

A self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis with real eigenvalues (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).

For a matrix Lie group the Lie-group exponential is the matrix exponential $e^X={\sum }_{k\ge 0}{X}^k\mathrm{/}k!$ (Matrix exponential as the Lie-group exponential).

If $G$ is a connected real semisimple Lie group with finite center and $K=G^{\mathrm{\Theta }}$ for a global Cartan involution with differential $\theta$ that fixes $Z(G)$ pointwise, then $K\times {\mathfrak{p}}_0\to G$, $(k,X)\mapsto k\exp X$, is a diffeomorphism (Global Cartan decomposition for a connected finite center semisimple Lie group, Cartan decomposition of a real semisimple Lie algebra).

Determinant is multiplicative, $\det (A^{-1})=(\det A{)}^{-1}$, and $\det A\mathrm{\ne }0$ exactly when $A$ is invertible (Determinant multiplicativity follows from the top exterior power, If $A$ is invertible over a commutative ring, then $\det (A^{-1})=\det (A{)}^{-1}$, For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix).

The real algebra ${\mathfrak{s}\mathfrak{l}}_n(\mathbb{R})$ is semisimple and $\theta (X)=-X^{\mathsf{T}}$ is its Cartan involution with symmetric traceless anti-fixed space (Cartan involution and k plus p for sl n r, Cartan decomposition of a real semisimple Lie algebra).

Given: ${\mathfrak{g}}_0={\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ with the basis $e=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, $f=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$, $H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ satisfying $[H,e]=2e$, $[H,f]=-2f$, $[e,f]=H$, and the matrices $K_0=f-e=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $\theta$.

${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ consists of the real traceless $2\times 2$ matrices, with basis $e,f,H$ and the displayed bracket relations (The special linear Lie algebra sl_2, General and special linear Lie groups).

The Cartan involution $\theta (X)=-X^{\mathsf{T}}$ of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ has ${\mathfrak{k}}_0=\{X:X^{\mathsf{T}}+X=0\}=\mathfrak{s}\mathfrak{o}(2)$ and ${\mathfrak{p}}_0=\{X\in {\mathfrak{s}\mathfrak{l}}_2(\mathbb{R}):X^{\mathsf{T}}=X\}$ (Cartan involution and k plus p for sl n r, Cartan involution of a real semisimple Lie algebra, Cartan decomposition of a real semisimple Lie algebra, Orthogonal and special orthogonal Lie groups).

A Cartan subalgebra of a Lie algebra is a nilpotent self-normalizing subalgebra, and a $\theta$-stable Cartan subalgebra ${\mathfrak{h}}_0$ decomposes as ${\mathfrak{h}}_0={\mathfrak{t}}_0\oplus {\mathfrak{a}}_0$ with ${\mathfrak{t}}_0={\mathfrak{h}}_0\cap {\mathfrak{k}}_0$ the compact part and ${\mathfrak{a}}_0={\mathfrak{h}}_0\cap {\mathfrak{p}}_0$ the split part; it is maximally compact when $\dim {\mathfrak{t}}_0$ is maximal and maximally noncompact when $\dim {\mathfrak{a}}_0$ is maximal (Cartan subalgebra, Theta-stable Cartan subalgebras and their compact and split parts).

Proof technique: direct matrix computation.

1.1 The elements $K_0$ and $H$ lie where claimed: $K_0^{\mathsf{T}}=-K_0$, so $\theta K_0=K_0$ and $K_0\in {\mathfrak{k}}_0$, while $H^{\mathsf{T}}=H$ and $\mathrm{tr}H=0$, so $\theta H=-H$ and $H\in {\mathfrak{p}}_0$. In particular both lines $\mathbb{R}{K}_0$ and $\mathbb{R}H$ are $\theta$-stable. [given, L2, algebra]

1.2 The normalizer of $\mathbb{R}{K}_0$ in ${\mathfrak{g}}_0$ is $\mathbb{R}{K}_0$: for $X=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$ one computes $[K_0,X]=\left(\begin{array}{cc}-b-c& 2a\\ 2a& b+c\end{array}\right)$. For this matrix to equal $r{K}_0$ its diagonal entries force $c=-b$, while its two off-diagonal entries give $2a=-r$ and $2a=r$, hence $a=r=0$. Thus $X\in \mathbb{R}{K}_0$, and conversely every such $X$ normalizes the line. Hence $\mathbb{R}{K}_0$ is abelian, nilpotent and self-normalizing, so it is a Cartan subalgebra by [L3]. [given, L1, L3, algebra]

1.3 The normalizer of $\mathbb{R}H$ in ${\mathfrak{g}}_0$ is $\mathbb{R}H$: for $X=\left(\begin{array}{cc}a& b\\ c& -a\end{array}\right)$ one computes $[H,X]=\left(\begin{array}{cc}0& 2b\\ -2c& 0\end{array}\right)$. If this equals $rH$, comparison of diagonal and off-diagonal entries gives $r=0$ and $b=c=0$, so $X=\mathrm{diag}(a,-a)\in \mathbb{R}H$; conversely every such $X$ normalizes the line. Hence $\mathbb{R}H$ is abelian, nilpotent and self-normalizing, so it is a Cartan subalgebra by [L3]. [given, L1, L3, algebra]

1.4 The adjoint spectra distinguish the two: by [L1], $[K_0,H]=[f-e,H]=[f,H]-[e,H]=2f+2e=2(e+f)$ and $[K_0,e+f]=[f,e]+[f,f]-[e,e]-[e,f]=[f,e]-[e,f]=-2H$, so ${\mathrm{ad}}_{K_0}$ is zero on $K_0$ and has the matrix $\left(\begin{array}{cc}0& -2\\ 2& 0\end{array}\right)$ in the basis $(H,e+f)$ of ${\mathfrak{p}}_0$, with eigenvalues $0,\pm 2i$; whereas ${\mathrm{ad}}_H$ is diagonal on $e,f,H$ with real eigenvalues $2,-2,0$. [given, L1, algebra]

2.1 Both Cartan subalgebras are $\theta$-stable by step 1.1, so the decomposition of [L3] applies. For ${\mathfrak{h}}_0=\mathbb{R}{K}_0$ one has ${\mathfrak{h}}_0\cap {\mathfrak{k}}_0=\mathbb{R}{K}_0$ and ${\mathfrak{h}}_0\cap {\mathfrak{p}}_0=0$, so ${\mathfrak{t}}_0={\mathfrak{h}}_0$ and ${\mathfrak{a}}_0=0$: every element of ${\mathfrak{h}}_0$ is compact, and the compact dimension $1$ equals $\dim {\mathfrak{k}}_0$, so ${\mathfrak{h}}_0$ is maximally compact. For ${\mathfrak{h}}_0^{\mathrm{\prime }}=\mathbb{R}H$ one has ${\mathfrak{h}}_0^{\mathrm{\prime }}\cap {\mathfrak{k}}_0=0$ and ${\mathfrak{h}}_0^{\mathrm{\prime }}\cap {\mathfrak{p}}_0=\mathbb{R}H$, so ${\mathfrak{t}}_0=0$ and ${\mathfrak{a}}_0={\mathfrak{h}}_0^{\mathrm{\prime }}$. Its noncompact dimension is maximal: if a $\theta$-stable Cartan subalgebra $\mathfrak{q}$ had $\dim (\mathfrak{q}\cap {\mathfrak{p}}_0)=2$, then ${\mathfrak{p}}_0=\mathbb{R}H\oplus \mathbb{R}(e+f)$ would lie in $\mathfrak{q}$. Closure under brackets and $[H,e+f]=2(e-f)=-2K_0$ would then put $K_0$ in $\mathfrak{q}$, hence $\mathfrak{q}={\mathfrak{g}}_0$. But ${\mathfrak{g}}_0$ is not nilpotent (indeed $[{\mathfrak{g}}_0,{\mathfrak{g}}_0]={\mathfrak{g}}_0$ from the displayed basis relations), whereas every Cartan subalgebra is nilpotent by [L3]. Thus every split part has dimension at most $1$, and $\mathbb{R}H$ attains that bound. [step 1.1, step 1.2, step 1.3, L1, L2, L3, algebra]

3.1 Consistency of brackets and signs with the general theory: $[{\mathfrak{h}}_0,{\mathfrak{h}}_0]=0$ lies in both ${\mathfrak{t}}_0$ and ${\mathfrak{a}}_0$ cases because the Cartan subalgebras are abelian, and the compact line lies in ${\mathfrak{k}}_0$ where the Killing form is negative definite while the split line lies in ${\mathfrak{p}}_0$ where it is positive definite, so the Killing form is negative definite on ${\mathfrak{t}}_0={\mathfrak{h}}_0$ in the first case and positive definite on ${\mathfrak{a}}_0={\mathfrak{h}}_0^{\mathrm{\prime }}$ in the second. Both subalgebras are one-dimensional, and the two eigenvalue computations of step 1.4 match the compact/purely imaginary and split/real terminology. [step 1.1, step 1.4, step 2.1, L2, algebra]

4.1 Endpoints and scope: the algebra is three-dimensional and both Cartan subalgebras are one-dimensional, which equals its rank; ${\mathfrak{h}}_0\mathrm{\ne }{\mathfrak{h}}_0^{\mathrm{\prime }}$, and the compact one is the compact line $\mathbb{R}{K}_0=i\mathbb{R}{h}_B$ in the notation of the source and the split one is the diagonal line $\mathbb{R}H$. No choice principle enters, and the computations are finite. [given, step 1.2, step 1.3, step 2.1, algebra] ∎

Given: The Axiom of Choice; the group ${\mathrm{SL}}_2(\mathbb{R})$ of real $2\times 2$ matrices of determinant $1$; the Cartan involution $\theta (X)=-X^{\mathsf{T}}$ with ${\mathfrak{k}}_0=\mathfrak{s}\mathfrak{o}(2)$ and ${\mathfrak{p}}_0$ the symmetric traceless matrices; and a matrix $g=\left(\begin{array}{cc}p& r\\ q& s\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{R})$.

The Axiom of Choice is The Axiom of Choice; it enters only through the global Iwasawa theorem of [L4] and the smooth structure of the group.

${\mathrm{SL}}_2(\mathbb{R})$ is an embedded Lie group with Lie algebra ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ and $\mathrm{SO}(2)$ is a closed connected subgroup with Lie algebra $\mathfrak{s}\mathfrak{o}(2)=\mathbb{R}\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ (General and special linear Lie groups, Orthogonal and special orthogonal Lie groups).

The Cartan decomposition ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})={\mathfrak{k}}_0\oplus {\mathfrak{p}}_0$ has ${\mathfrak{k}}_0=\mathfrak{s}\mathfrak{o}(2)$ and ${\mathfrak{p}}_0=\{X:X^{\mathsf{T}}=X,\mathrm{tr}X=0\}$; ${\mathfrak{p}}_0$ is spanned by $H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ and $\sigma =\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, and $[H,\sigma ]=\left(\begin{array}{cc}0& 2\\ -2& 0\end{array}\right)\mathrm{\ne }0$, so $\mathbb{R}H$ is a maximal abelian subspace of ${\mathfrak{p}}_0$ (Cartan involution and k plus p for sl n r, Compact and split cartan subalgebras of sl two r, Cartan decomposition of a real semisimple Lie algebra, Theta-stable Cartan subalgebras and their compact and split parts).