Dot Product of Vector Cross Products/Proof 2

Proof
Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be embedded in a Cartesian space:

where $\tuple {\mathbf e_1, \mathbf e_2, \mathbf e_3}$ denotes the standard ordered basis of $\mathbf V$.

Then:

Hence the result.