Exercise 3, § 2

Let G be a compact Lie group, $\mathfrak{g}$ its Lie algebra, $\mathfrak{s}$ a vector subspace of $\mathfrak{g}$ such that, for $x, y, z$ in $\mathfrak{s}$, we have $[x,[y, z]]\in \mathfrak{s}$. Let $\mathscr{L}$ be the set of commutative subalgebras of $\mathfrak{g}$ contained in $\mathfrak{s}$. Show that the identity component of the stabilizer of $\mathfrak{s}$ in G operates transitively on the set of maximal elements of $\mathscr{L}$ (argue as in the proof of Th. 1).