Scalar triple product squared identity

The scalar triple product is [a b c] = a · (b × c), and it is unchanged by cyclic permutation: [a b c] = [b c a] = [c a b].

We need the identity P × (Q × R) = Q(P · R) − R(P · Q).

Step 1: Simplify (c × a) × (a × b)

Take P = c × a, Q = a, R = b:

(c × a) × (a × b) = a[(c × a) · b] − b[(c × a) · a]

Now (c × a) · b = [c a b] = [a b c], and (c × a) · a = 0 (a cross product is perpendicular to its factors). So

(c × a) × (a × b) = [a b c] a.

Step 2: Evaluate the triple product

[b×c c×a a×b] = (b × c) · [(c × a) × (a × b)]

= (b × c) · [a b c] a
= [a b c] (b × c) · a
= [a b c] [a b c]

because a · (b × c) = [a b c].

Hence [b×c c×a a×b] = [a b c]².

(Numerical check with a = i, b = j, c = k: b×c = i, c×a = j, a×b = k, so the left side is [i j k] = 1 and [a b c]² = 1.)