BAC-CAB rule | vector identities

The vector triple product identity Ax(BxC) = B(A·C) - C(A·B). This identity can be generalized to n dimensions, a_2x...xa_(n - 1)x(b_1x...xb_(n - 1)) = (-1)^(n + 1) left bracketing bar b_1 | ... | b_(n - 1) a_2·b_1 | ... | a_2·b_(n - 1) ⋮ | ⋱ | ⋮ a_(n - 1)·b_1 | ... | a_(n - 1)·b_(n - 1) right bracketing bar .