Two nonconjugate real cartan subalgebras

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

${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ is the real Lie algebra of real traceless matrices; its inner automorphisms are the maps ${\mathrm{Ad}}_g(X)=gX{g}^{-1}$ for $g\in {\mathrm{SL}}_2(\mathbb{R})$, and more generally every automorphism $\phi$ satisfies $\phi ({\mathrm{ad}}_H)={\mathrm{ad}}_{\phi (H)}\phi$ (General and special linear Lie groups, The special linear Lie algebra sl_2).

A Cartan subalgebra is a nilpotent self-normalizing subalgebra (Cartan subalgebra).

The lines $\mathbb{R}{K}_0$ and $\mathbb{R}H$ are $\theta$-stable Cartan subalgebras of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$: $\mathbb{R}{K}_0$ is the compact one and $\mathbb{R}H$ the split one (Compact and split cartan subalgebras of sl two r, Cartan involution and k plus p for sl n r).

The two Cartan subalgebras $\mathbb{R}H$ and $\mathbb{R}{K}_0$ of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ are not conjugate by any real inner automorphism (Real Cartan subalgebras need not be conjugate).

Proof technique: direct computation of adjoint spectra.

1.1 The subspaces $\mathbb{R}{K}_0$ and $\mathbb{R}H$ are Cartan subalgebras of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ by [L3], and they are distinct, because $K_0$ is skew-symmetric while $H$ is symmetric and diagonal. [given, L3, algebra]

1.2 The adjoint operator of $K_0$ has spectrum $\{0,2i,-2i\}$: from the relations one computes $[K_0,H]=[f-e,H]=2f+2e=2(e+f)$ and $[K_0,e+f]=[f,e]-[e,f]=-H-H=-2H$, while $[K_0,K_0]=0$; hence in the basis $(H,e+f,K_0)$ of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ the operator ${\mathrm{ad}}_{K_0}$ has the block matrix $\left(\begin{array}{ccc}0& -2& 0\\ 2& 0& 0\\ 0& 0& 0\end{array}\right)$, whose characteristic polynomial is $t(t^2+4)$. [given, algebra]

1.3 The adjoint operator of $H$ has spectrum $\{0,2,-2\}$: by the given relations ${\mathrm{ad}}_H$ is diagonal in the basis $(H,e,f)$ with eigenvalues $0,2,-2$, so its characteristic polynomial is $t(t-2)(t+2)$. [given, algebra]

2.1 No automorphism of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ carries $\mathbb{R}H$ onto $\mathbb{R}{K}_0$: if $\phi$ were such an automorphism with $\phi (H)=c{K}_0$ for some $c\mathrm{\ne }0$, then by [L1] the operators ${\mathrm{ad}}_{\phi (H)}$ and ${\mathrm{ad}}_H$ would be conjugate, hence would have the same characteristic polynomial; but step 1.3 gives $t(t-2)(t+2)$ for ${\mathrm{ad}}_H$ and step 1.2 gives $t(t^2+c^2\cdot 4)$ for ${\mathrm{ad}}_{c{K}_0}$, and no nonzero $c$ makes these polynomials equal (the first has three distinct real roots, the second has a nonzero purely imaginary pair). [step 1.2, step 1.3, L1, algebra]

3.1 Consequently the two Cartan subalgebras $\mathbb{R}H$ and $\mathbb{R}{K}_0$ are not conjugate by any automorphism, and in particular not by an inner automorphism; since they are distinct Cartan subalgebras of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$ by step 1.1, they refute the displayed statement, and they are exactly a witness pair for the general phenomenon of [L4]. [step 1.1, step 2.1, L4]

4.1 Scope: the invariant that separates the two lines is the isomorphism type of ${\mathrm{ad}}_H$ as a real operator, equivalently the position of the line inside ${\mathfrak{k}}_0$ or ${\mathfrak{p}}_0$: the compact line consists of elements whose adjoint operators have purely imaginary nonzero spectrum, the split line of elements with real nonzero spectrum. The computation is finite, uses no choice principle, and shows that the failure of conjugacy is detected already at the level of all automorphisms, not merely inner ones. [step 1.2, step 1.3, step 2.1, algebra] ∎