Tame Automorphisms of Free Nilpotent Lie Algebras of Countably Infinite Rank

Let $L_{\mathrm{\infty }}$ be a free Lie algebra of countably infinite rank over a field of characteristic $0$ and let $L_{\mathrm{\infty },c}$ be the free nilpotent Lie algebra of countably infinite rank and class $c$. We prove that every automorphism of $L_{\mathrm{\infty },c}$ is induced by an automorphism of $L_{\mathrm{\infty }}$.