Exercise 15, § 2

Let $X = \{x, y\}$ be a set with two elements, and let $\sigma$ be an automorphism of the Lie algebra $L(X)$. Now prove that, if $K$ is a field, $\sigma$ respects the grading of $L(X)$ (apply Exercise 14 to the non-zero homogeneous components of highest degree of $\sigma(x)$ and $\sigma(y)$); 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 square zero, prove that $x \mapsto x, y \mapsto y + \varepsilon[x, y]$ extends to an automorphism of $L(X)$ which does not respect the grading of $L(X)$.