Corollary

Let E be a complete normed space over $\mathbf{C}$, $v \in \mathscr{L}(E)$ and $g = \exp v$. Suppose that every $z \in \mathrm{Sp}\, v$ satisfies $-\frac{\pi}{2} < \mathscr{J}(z) < \frac{\pi}{2}$. Then, for every closed vector subspace M of $\mathscr{L}(E)$, the conditions $gMg^{-1} = M$ and $[v, M] \subset M$ are equivalent.

Let $F = \mathscr{L}(E)$, $g'$ be the mapping $f \mapsto gfg^{-1}$ of F into F and $v'$ be the mapping $f \mapsto [v, f]$ of F into F. Then $g' = \exp v'$ (no. 4, Corollary 3 to Proposition 10 and § 3, no. 11, Corollary 1 to Proposition 41). Lemma 6 proves that $-\pi < \mathscr{J}(z) < \pi$ for all $z \in \mathrm{Sp}\, v'$. It then suffices to apply Proposition 19.