problem understanding $a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$

Question

$a\times(b\times(a\times b))=a\times((b\cdot b)a-(b\cdot a)b)=-a\cdot ba\times b$
can anyone expand on how the final answer is derived?
I try to expand but ended up scratching my head.
the best I can do is this
$a\times(ab\cdot b - ab\cdot b)$

Answer 1

In $a\times((b\cdot b)a-(b\cdot a)b)$, the first term is zero because $a \times a = 0$. This leaves the second term. Note that $a\cdot b = b\cdot a$.