Let $u_1$ and $u_2$ be orthonormal vectors in an inner product space $U=(U,\langle\cdot,\cdot\rangle)$ over $F$ and let $a$ be in $F$. Show that the linear transformation $P_u = \langle u,u_1 + au_2\rangle u_1$ is a projection but not orthogonal unless $a = 0$.
I know that orthonormal implies the inner product of u1 and u2 is zero, and that each has unit length. I need to show P^2 = P, and I know this should be trivial but I don't understand properties of the inner product. I have [u + u1,u2] = [u,u2] + [u1,u2] and other relations from the definition of inner product, but I can't seem to string them together to show Pu = P(Pu).