Vector Cross Product of Vector Cross Products

Theorem
Let $\mathbf a, \mathbf b, \mathbf c, \mathbf d$ be vectors in a vector space $\mathbf V$ of $3$ dimensions:

where $\left({\mathbf e_1, \mathbf e_2, \mathbf e_3}\right)$ is the standard ordered basis of $\mathbf V$.

Let $\mathbf a \times \mathbf b$ denote the vector cross product of $\mathbf a$ with $\mathbf b$.

Let $\mathbf a \cdot \mathbf b$ denote the dot product of $\mathbf a$ with $\mathbf b$.

Then: