Tomihisa's Identity

Tomihisa's identity is the relation

		(1)

for any five real three-dimensional vectors , , , , and , where denotes the cross product (Aicardi 2010, p. 15). It also holds with the Poisson bracket on binary quadratic forms (Tomihisa 2009; Aicardi 2010, p. 15).

To prove the cross product identity, write for the scalar triple product, and put . The cofactor identity

(2)

holds for every vector , including when . Let denote the left side of (1). Expanding its outer cross products using the vector triple product and applying the cofactor identity with and gives

			(3)
			(4)