What is the generalized form of this identity and how to interpret it?

Question

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.

Answer 1

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

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You could try to remember it "in parts"

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