Question on lifting the Weyl group into the group of inner automorphisms of $\mathfrak{g}$

Question

I'm looking for some clarification of a statement that I found in Kac and Peterson's paper (112 realizations of the basic representation of the loop group of $E_8$). Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra and $\mathfrak{h}$ a Cartan subalgebra. The Weyl group of $\mathfrak{g}$ acts on $\mathfrak{h}$. So, fix an element $w$ in the Weyl group and let $\mathfrak{h}_0$ be its fixed point set. The statement is, essentially, that there exists an $x \in \mathfrak{g}$ (not unique) such that:

\begin{equation} (1) \;\mathrm{Ad} \; \mathrm{exp} (2 \pi i \; x) |_{\mathfrak{h}} = w, \quad \text{and} \quad (2) \; [x,\mathfrak{h}_0] = 0. \end{equation}

The existence of an $x \in \mathfrak{g}$ satisfying (1) is clear to me (Humphreys describes the process of lifting the simple reflections of the Weyl group into the group of inner automorphisms of $\mathfrak{g}$, and we can simply extend to an arbitrary element $w$). However, it is less obvious to me that one can find such an $x$ that also satisfies (2). This seems to be a well-known fact, but I've not been able to find a reference for it. Any help would be greatly appreciated.

Answer 1

The key result is this:

If $\mathfrak{h}_0 \subset \mathfrak{h}$ is the fixed subspace of $w \in W$, then $w$ (considered as a transformation on $\mathfrak{h}^*$ via the Killing form) can be expressed as a product of simple reflections $w_\alpha$, where $\alpha \in \mathfrak{h}^*$ is a root of $\mathfrak{g}$ and $\alpha(\mathfrak{h}_0) = 0$.

For a reference, see e.g. Carter, Simple Groups of Lie Type, Theorem 2.5.5 (rephrased somewhat to fit the current context).

Then the argument goes as follows: For any root $\alpha$, you can lift a simple reflection $w_\alpha$ to an $x \in X_{\alpha} \oplus X_{-\alpha}$ satisfying your condition (1). Here $X_{\alpha}$ is the subspace of $\mathfrak{g}$ for the root $\alpha$ in the Cartan decomposition. The property $\alpha(\mathfrak{h}_0) = 0$ then gives condition (2): $[x,\mathfrak{h}_0] = 0$. For a general product of reflections, take a Lie bracket of the corresponding $x$'s.