Exterior square of a direct sum
= Exterior square of a direct sum
{title2=$\bigwedge^2(V\oplus W)\cong\bigwedge^2V\oplus\bigwedge^2W\oplus(V\otimes W)$}
The <exterior square> of a <direct sum> splits into wedges of two vectors from $V$, two from $W$, or one from each. The mixed map sends $v\otimes w$ to $(v,0)\wedge(0,w)$. A combined <basis> proves it is an isomorphism, and the <exterior-power Lie algebra representation> proves equivariance for modules. No division by $2$ is used.