Bilinearity and alternatingness of
$$
[(x,v),(y,w)]=([x,y],xw-yv)
$$
are immediate. For three elements, the $\mathfrak g$ component of the Jacobi sum vanishes by the <Jacobi identity> in $\mathfrak g$. In the $V$ component, the coefficient of a vector such as $u$ is
$$
[x,y]u-x(yu)+y(xu)=0
$$
because the action is a <Lie algebra representation>; the other terms cancel cyclically in the same way. Hence the bracket satisfies Jacobi and defines the <semidirect product of a Lie algebra and a module> $\mathfrak g\ltimes V$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 102 / 2 / c / i / Solution

{title2=$\mathfrak g\ltimes V$}

For a <Lie algebra representation> of $\mathfrak g$ on $V$, the semidirect product $\mathfrak g\ltimes V$ is the vector space $\mathfrak g\oplus V$ with bracket
$$
[(x,v),(y,w)]=([x,y],xw-yv).
$$
The subspace $V$ is an abelian <ideal of a Lie algebra>.

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