Solution

Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 4 / 4 / i / Solution

{title2=$\operatorname{ad}\mathfrak g\subseteq\operatorname{Der}(\mathfrak g)$}

= Inner derivations of a Lie algebra
{synonym}

An inner <derivation of a Lie algebra> is $\operatorname{ad}_x:y\mapsto[x,y]$. The <Jacobi identity> makes this a <derivation of a Lie algebra>. For every <derivation of a Lie algebra> $D$, $[D,\operatorname{ad}_x]=\operatorname{ad}_{Dx}$, so the inner derivations form an <ideal of a Lie algebra> in the <derivation Lie algebra>. The <kernel> of $x\mapsto\operatorname{ad}_x$ is the <center of a Lie algebra>.

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