The following result is proved: Let A be a commutative real Banach algebra with unit 1. Let G denote the group of invertible elements of A and let G1 be the connected component of G containing 1. If the quotient group G/G1 contains an element of finite order other than G1, then the order of such an element must be 2. If the group G/G1 is of finite order, then its order must be 2n for some nonnegative integer n. © 2014 University of Houston.