I have a problem understanding the proof of Proposition 8.20, page 211, in Besse's Einstein Manifolds. He considers a semi-simple Lie algebra $\mathfrak{g}$ with $\operatorname{Ad}$-invariant scalar product, its adjoint group $G$, and an orbit $M$ in $\mathfrak{g}$ under the adjoint representation. By $\operatorname{Ad}$-invariance the Lie algebra decomposes into an orthogonal direct sum $$\mathfrak{g}=\ker\operatorname{ad}_w\oplus\operatorname{im}\operatorname{ad}_w\equiv L_w\oplus M_w\quad\forall w\in M.$$ $L_w$ is the Lie algebra of the stabilizer group $G_w$.
Proposition. The stabilizer group $G_w$ is the commutator group $C(S_w)$ of its connected center $S_w$.
Proof. The Lie algebra $\mathfrak{s}_w$ of the connected center $S_w$ of $G_w$ is the center of $L_w$. Now, since $\operatorname{ad}_w$ has no kernel on $M_w$, we infer that $L_w$ is exactly the commutator of $\mathfrak{s}_w$ in $\mathfrak{g}$, so that the commutator group of $S_w$ is exactly the connected component of the identity of $G_w$, that is $G_w$ itself.
What is meant by "$L_w$ is exactly the commutator of $\mathfrak{s}_w$ in $\mathfrak{g}$"? (Usually the commutator of $\mathfrak{s}_w$ is the Lie sub-algebra $[\mathfrak{s}_w,\mathfrak{s}_w]$ generated by all commutators in $\mathfrak{s}_w$, but this definition does not make sense here.)