Vector Cross Product of Vector Cross Products

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

where $\left({\mathbf i, \mathbf j, \mathbf k}\right)$ is the standard ordered basis of the vector space in question.

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: