= Perfect semidirect product criterion
{title2=$[\mathfrak g\ltimes V,\mathfrak g\ltimes V]=[\mathfrak g,\mathfrak g]\oplus\mathfrak gV$}
For a <Lie algebra representation> of $\mathfrak g$ on $V$, let $\mathfrak gV$ denote the <linear span> of $xv$ for $x\in\mathfrak g$ and $v\in V$. The <derived algebra> of the <semidirect product of a Lie algebra and a module> is the displayed <direct sum of vector spaces>. Every <Lie bracket> has first component in $[\mathfrak g,\mathfrak g]$ and second component in $\mathfrak gV$. Conversely, brackets of $(x,0)$ with $(y,0)$ span the former component, and brackets of $(x,0)$ with $(0,v)$ span the latter. Therefore this <semidirect product of a Lie algebra and a module> is a <perfect Lie algebra> exactly when $\mathfrak g$ is a <perfect Lie algebra> and $\mathfrak gV=V$. When $V\ne0$, its nonzero <abelian Lie algebra> ideal prevents it from being a <semisimple Lie algebra>. The <sl2 Lie algebra> acting on its two-dimensional <defining representation> supplies a five-dimensional example.
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