Vector Cross Product of Vector Cross Products/Corollary

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

Let $\sqbrk {\mathbf a, \mathbf b, \mathbf c}$ denote the scalar triple product of $\mathbf a$, $\mathbf b$ and $\mathbf c$.

Then:
 * $\sqbrk {\mathbf a, \mathbf b, \mathbf c} \mathbf d = \sqbrk {\mathbf b, \mathbf c, \mathbf d} \mathbf a + \sqbrk {\mathbf a, \mathbf d, \mathbf c} \mathbf b + \sqbrk {\mathbf a, \mathbf b, \mathbf d} \mathbf c$

Proof
From Vector Cross Product of Vector Cross Products: