Corollary 2

Let E be a complete normable algebra over $\mathbf{C}$ and $w \in \mathscr{L}(E)$. Suppose that every $z \in \mathrm{Sp}\,w$ satisfies $|\mathscr{J}(z)| < \frac{2\pi}{3}$. The following conditions are equivalent: (i) w is a derivation of E; (ii) exp w is an automorphism of E. This follows from Corollary 1 with $n = 2$ and $f$ the multiplication of E.