On some orthogonally additive functions on inner product spaces

Abstract. Let \(E\) be a real inner product space of dimension at least 2. If \( f: E \to E \) satisfies $$ f(x+y) = f(x) + f(y) \quad \textrm{for all orthogonal } x, y \in E $$ and $$ f(f(x)) = x \quad \textrm{for } x \in E, $$ then \(f\) is additive.