= Exterior square of the defining orthogonal representation
{title2=$\Lambda^2 V\cong\mathfrak{so}(V)$}
For a complex <vector space> with a <nondegenerate> <symmetric bilinear form> $Q$, the map $u\wedge v\mapsto[w\mapsto Q(v,w)u-Q(u,w)v]$ identifies the <exterior square> of the defining representation with the <Adjoint representation of a Lie algebra> of $\mathfrak{so}(V)$. Equivariance follows from invariance of $Q$; an orthogonal basis verifies bijectivity. It is irreducible when $\mathfrak{so}(V)$ is simple. In dimension four it splits into the adjoint modules of the two commuting <sl2 Lie algebras>.
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