called the the automorphism Lie algebra of $\mathfrak{g}$ (or derivation Lie algebra), is the Lie algebra whose underlying vector space is that of those linear maps $\Delta \colon \mathfrak{g} \to \mathfrak{g}$ which satisfy the derivation property:

$\Delta([x,y]) = [\Delta(x), y] + [x, \Delta(y)]$

for all $x,y \in \mathfrak{g}$. The Lie bracket on $\mathfrak{aut}(\mathfrak{g})_{\mathrm{even}}$ is the commutator operation: