= Exterior square realization of the orthogonal adjoint representation
{title2=$u\wedge v\mapsto(w\mapsto\langle v,w\rangle u-\langle u,w\rangle v)$}
For a real <inner product space> $V$, the displayed map identifies $\Lambda^2V$ with its <Special orthogonal Lie algebra>. It sends a wedge of distinct orthonormal basis vectors to the corresponding elementary <skew-symmetric matrix>, so it is an isomorphism. For an <orthogonal transformation> $g$, invariance of the <inner product> gives $\Phi(gu\wedge gv)=g\Phi(u\wedge v)g^{-1}$. Thus the <exterior square> of the defining representation is the <Adjoint representation> of the orthogonal group. Complexification identifies their <weight spaces>, allowing the <root system> to be read from the exterior-square weights and the zero-weight multiplicity.
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