Function that Recovers the Inner Product

For $\mathbb{R}$

$⟨\alpha |\beta ⟩=\frac{1}{4}||\alpha +\beta |{|}^2-\frac{1}{4}||\alpha -\beta |{|}^2$

For $\mathbb{C}$

$⟨\alpha |\beta ⟩=\frac{1}{4}||\alpha -\beta |{|}^2-\frac{1}{4}||\alpha -\beta |{|}^2+i\frac{1}{4}||\alpha +i\beta |{|}^2-i\frac{1}{4}||\alpha -i\beta |{|}^2$