Exterior square of a direct sum (${\bigwedge }^2(V\oplus W)\cong {\bigwedge }^{2}V\oplus {\bigwedge }^{2}W\oplus (V\otimes W)$)

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.