A Universal Identity for Powers in Quadratic Algebras and a Matrix Derivation of a Fibonacci Identity

We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant. Applying this to the Fibonacci matrix yields a binomial expansion formula for F_{nm}, recovering a recent identity of Vorobtsov. This shows that such identities arise from general algebraic principles rather than specific properties of Fibonacci numbers.