Let $\theta$ and $\tau$ be involutions of a semisimple complex lie algebra $\mathfrak{g}$, and suppose that they have the same Satake diagram. I want to understand why they must be conjugate to one another.
Let $\mathfrak{t}_-$ be a maximal torus in the $(-1)$-eigenspace of $\theta$, and let $\mathfrak{h}$ be a Cartan subalgebra containing $\mathfrak{t}_1$. Then I see that we can conjugate $\tau$ by an automorphism $\sigma$ so that $\sigma^{-1}\tau\sigma|_{\mathfrak{h}}=\theta|_{\mathfrak{h}}$. It will then follow that $(\sigma^{-1}\tau\sigma)\circ\theta$ fixes $\mathfrak{h}$ and further fixes the centralizer of $\mathfrak{t}_1$ (since both $\theta$ and $\sigma^{-1}\tau\sigma$ fix it individually). But this doesn't imply it's the identity map, of course, or even that it's conjugate to the identity (since there are many automorphisms that fix a Cartan pointwise without being conjugate to the identity).
So I'm stuck there. Any help/ideas?
Edit: For clarification, by involution of $\mathfrak{g}$ I mean an automorphism of $\mathfrak{g}$ as a complex lie algebra which is not the identity but composes with itself to the identity.
Satake Diagram: if $\theta$ is an involution of $\mathfrak{g}$, then we can write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$, where $\mathfrak{k}$ is the (+1)-eigenspace of $\theta$ and $\mathfrak{p}$ is the $(-1)$-eigenspace of $\theta$. Choosing a maximal torus $\mathfrak{t}_1\subseteq\mathfrak{p}$ (i.e. a maximal abelian subspace consisting of semisimple elements inside $\mathfrak{p}$), we can extend this to a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$, and choose a Borel by taking a regular element $\gamma\in\mathfrak{t}_1^*$ and perturbing it slightly to a regular element in $\mathfrak{h}^*$. For such a choice of Cartan subalgebra and Borel, we take the corresponding Dynkin diagram from simple positive roots $\{\alpha_1,\dots,\alpha_n\}$ and do the following: if $\alpha_i|_{\mathfrak{t}_1}=0$, color the node for $\alpha_i$ black. If $\alpha_i|_{\mathfrak{t}_1}\neq0$ color the node for $\alpha_i$ white. If $\alpha_j|_{\mathfrak{t}_1}=\alpha_k|_{\mathfrak{t}_1}\neq 0$, then connect the nodes for $\alpha_i$ and $\alpha_j$ by arrows. The resulting diagram is called the 'Satake diagram' of $\theta$, and is well-defined.
Reference: 'Homogeneous Spaces and Equivariant Embeddings' by D.A. Timashev.
