Proving [b x c, c x a, a x b] = [a, b, c]^2 with Vector Identity Proof

[Lunat1c]

Homework Statement

Hi. I need to prove that [b x c, c x a, a x b] = [a, b, c]² for any three vectors a, b and c.

Note that [a, b, c] = a(b x c)

Homework Equations

I tried using the identify (a x b) x c = (a.c)b - (a.b)c

The Attempt at a Solution

Using the above identity I got [b x c, c x a, a x b] = (b x c){(c x a) x (a x b)} = (b x c){(a(a x b)a - (ca)(a x b)}. I'm not sure where to go from here or if it's the correct approach at all

Thanks for any help!

 

[CompuChip]

If a, b, c are vectors, then (b x c) is a vector. So then, what does the multiplication a(b x c) mean? Is it inner product of a with (b x c)? Are the vectors three dimensional?
Or are you working in a Clifford algebra where ab is the Clifford (geometrical) product and x is the commutator product (b x c = 1/2(bc - cb)).