Let $H$ be a Hilbert space and $\sigma: H \times H \to \mathbb{C}$ a sesquilinear form (linear in the first variable, anti-linear in the second variable). Then it is well-known that the following polarization identity holds:
$$\sigma(x,y)=\frac{1}{4} \sum_{k=0}^3 i^k \sigma(x+i^k y, x + i^ky)$$
Is there a similar result when $\sigma: H \times H \to \mathbb{C}$ is bilinear (linear in both variables)? I.e. is there a way to express $\sigma(x,y)$ as a linear combination of elements of the form $\sigma(z,z)$?
It's even easier: \begin{align*} \sigma(x,y) &= \frac{\sigma(x+y,x+y)-\sigma(x,x)-\sigma(y,y)}{2} \\ &=\frac{\sigma(x+y,x+y)-\sigma(x-y,x-y)}{4}. \end{align*}