in 3 dimensions. For example, the direction of a × b in 7-dimensions may be the same as c × d even though c and d are linearly independent of a and b. Also the seven-dimensional cross product is not compatible with the Jacobi identity. Quaternions A quaternion p is defined as the sum of a scalar t and a vector v: The product of two quaternions and is defined by The quaternionic conjugate of q is defined by and the norm squared is The multiplicativity of the norm in the quaternion algebra provides, for quaternions p and q: The quaternions p and q are called imaginary if their scalar part is zero; equivalently, if Lagrange's identity is just the multiplicativity of the norm of imaginary quaternions, since, by definition, Proof of algebraic form The vector form follows from the Binet-Cauchy identity by setting ci = ai and di = bi. The second version follows by letting ci and di denote the complex conjugates of ai and bi, respectively, Here is also a direct proof. The expansion of the first term on the left side is: which means that the product of a column of as and a row of bs yields (a sum of elements of) a square of abs, which can be broken up into a diagonal and a pair of triangles on either side of the diagonal. The second term on the left side of Lagrange's identity can be expanded as: which means that a symmetric square can be broken up into its diagonal and a pair of equal triangles on either side of the diagonal. To expand the summation on the right side of Lagrange's identity, first expand the square within the summation: Distribute the summation on the right side, Now exchange the indices i and j of the second term on the right side, and permute the b factors of the third term, yielding: Back to the left side of Lagrange's identity: it has two terms, given in expanded form by Equations () and (). The first term on the right side of Equation () ends up canceling out the first term on the right side of Equation (), yielding which is the same as Equation (), so Lagrange's identity is indeed an identity, Q.E.D. Proof of Lagrange's identity for complex numbers Normed division algebras require that the norm of the product is equal to