Cross, Dot Product Properties - Volume of Brillouin Zone

Question

(See image) I'm trying to understand how the final line of this answer is reached. I know that an identity is used to get from the 3rd last line to the 2nd last line, but what properties of dot & cross products are used in the final step?

Thank you

Answer 1

First pull the scalar out:

$$ (a_2\times a_3)\cdot\color{Blue}{(a_3\cdot(a_1\times a_2))}a_1=\color{Blue}{(a_3\cdot(a_1\times a_2))}((a_2\times a_3)\cdot a_1) $$

Then use the fact the scalar triple product is cyclic to see both triple products above are identical.

Answer 2

Alternatively use $(a\times b)\cdot(c\times d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c)$ in$$\begin{align}(a_2\times a_3)\cdot(a_3\times a_1)\times(a_1\times a_2)&=(a_2\cdot a_3\times a_1)(a_3\cdot a_1\times a_2)\\&-(a_2\cdot a_1\times a_2)(a_3\cdot a_3\times a_1)\\&=(a_1\cdot a_2\times a_3)^2-0^2\\&=(a_1\cdot a_2\times a_3)^2\end{align}$$to get from the first line to the third.