What is the generalized form of this identity and how to interpret it? I have learnt that for any inner product space of $\mathbb{C}$, we have
$$\langle f,g\rangle=\frac{1}{4}\Big[||f+g||^2-||f-g||^2+i\big(||f+ig||^2-||f-ig||^2\big) \Big]$$
I know how to prove it, but I have a hard time to remember the formula. I think maybe there are some general form or tricks which can help me to remember the formula, or there may be some geometric interpretation of the identity. 

This was an answer to the original question, which specifically asked for help memorizing the identity.

You could try to remember it "in parts"


*

*$[f,g] = \|f+g\|^2-\|f-g\|^2$

*$\langle f,g \rangle = \frac{1}{4}([f,g]+i[f,ig])$


If someone can offer a geometric interpretation of $[f,g],$ that would certainly help.
