Peter J. Cameron's "Notes on Linear Algebra" defines
An inner product on a real vector space $V$ is a function $b: V \times V \to R$ satisfying
b is bilinear (that is, b is linear in the first variable when the second is kept constant and vice versa);
b is positive definite, that is, $b(v,v) \geq 0$ for all $v \in V$, and $b(v,v) = 0$ if and only if $v = 0$.
Using the notation $x \cdot y = b(x,y)$, it then expands (without explanation) $(v+xw) \cdot (v+xw)$ to $x^2 (w \cdot w) + 2x (v \cdot w) + v \cdot v$.
However when I try to verify this, I get to $x^2(w \cdot w) + x(v \cdot w) + x(w \cdot v) + v \cdot v$, which clearly equals the desired expression if b is symmetric. But I don't believe we've established anywhere that it is, so I guess I'm either misunderstanding a definition, or I'm missing a trick when performing the expansion.