the product of the norms. Lagrange's identity exhibits this equality. The product identity used as a starting point here, is a consequence of the norm of the product equality with the product of the norm for scator algebras. This proposal, originally presented in the context of a deformed Lorentz metric, is based on a transformation stemming from the product operation and magnitude definition in hyperbolic scator algebra. Lagrange's identity can be proved in a variety of ways. Most derivations use the identity as a starting point and prove in one way or another that the equality is true. In the present approach, Lagrange's identity is actually derived without assuming it a priori. Let be complex numbers and the overbar represents complex conjugate. The product identity reduces to the complex Lagrange's identity when fourth order terms, in a series expansion, are considered. In order to prove it, expand the product on the LHS of the product identity in terms of series up to fourth order. To this end, recall that products of the form can be expanded in terms of sums as where means terms with order three or higher in . The two factors on the RHS are also written in terms of series The product of this expression up to fourth order is Substitution of these two results in the product identity give The product of two conjugates series can be expressed as series involving the product of conjugate terms. The conjugate series product is thus The terms of the last two series on the LHS are grouped as in order to obtain the complex Lagrange's identity: In terms of the moduli, Lagrange's identity for complex numbers has been obtained from a straightforward product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, this proof is yet another way to obtain the CS inequality. Higher order terms in the series produce novel identities. See also Brahmagupta–Fibonacci identity Lagrange's identity (boundary value problem) Binet–Cauchy identity References External links Mathematical identities Multilinear algebra Articles containing proofs