Exercise 15, § 2

Let $X = \{x, y\}$ be a set with two elements and let $\sigma$ be an automorphism of the Lie algebra $L(X)$. Show that, if $K$ is a field, $\sigma$ preserves the graduation of $L(X)$ (apply Exercise 14 to the non-zero homogeneous components of $\sigma(x)$ and $\sigma(y)$ of highest degree); deduce an isomorphism of $\mathrm{Aut}(L(X))$ onto $\mathbf{GL}(2, K)$. Extend these results to the case where the ring $K$ has no nilpotent element $\neq 0$. When $K$ contains an element $\varepsilon \neq 0$ of zero square, show that $x \mapsto x, \ y \mapsto y + \varepsilon[x, y]$ can be extended to an automorphism of $L(X)$ which does not preserve the graduation of $L(X)$.