Cartan involution and k plus p for sl n r

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] ∎