Show true for all vectors

Use Maple to show that, for these particular vectors,
$(a \times b).[(b \times c) \times (c \times a)] = [a.(b \times c)]^2$ where $\times$ is the vector cross product.

I was able to do this but for the second part of the question it asks:

Using pen-and-paper, show that equation (z) holds for ALL vectors a, b and c
Would I just set vector $a$ as $\begin{pmatrix} a_1\\a_2\\a_3\end{pmatrix}$, $b$ as $\begin{pmatrix} b_1\\b_2\\b_3\end{pmatrix}$ etc and work through to get a big equation in the form of $abc$ in the end, or is there a more elegant way?

Thanks in advance

Craig