Kunita-Watanabe identity:
Let $M,N$ be local martingales, $H$ be a locally bounded previsible process, then $[H\cdot M,N]=H\cdot[M,N],$ where $[M,N]$ is covariation.
I am going though the proof, but 1 step is omitted. I am trying to use the polarization to conclude that $[H\cdot M,N]+[M,H\cdot N]=2H\cdot [M,N],$ i.e. $\frac{1}{4}([H\cdot M+N]-[H\cdot M-N]+[M+H\cdot N]-[M-H\cdot N])=2H\cdot [M,N],(*)$ where $[X]$ denotes quadratic variation of $X$. How can I show $(*)$?
Thank you.