Strongly orthogonal martingales Let $\mathscr{H}_0^2$ the space of all $L^2$ martingales $X$ starting at zero s.t. $\sup_{t\geq 0}E[X_t^2]<\infty$.
Two martingales $N,M$ are said to be strongly orthogonal, if $NM$ is a uniformly integrable martingale. 
In Protter, Philip: Stochastic Integration and Differential Equation, there is a characterization for strongly orthogonal martingales (in chapter IV.3 after the definition of strongly orthogonal martingales).

Let $N,M\in \mathscr{H}_0^2$. Then $N,M$ are strongly orthogonal iff $[N,M]$ is a uniformly integrable martingale.

$"\Rightarrow"$ By definition, $[N,M]=NM-N_{-}\cdot M-M_{-}\cdot N$. $NM$ is a martingale by assumption, but are $N\cdot M$ and $M\cdot N$ martingales (they are clearly local martingales)?
$[N,M]$ is the unique process s.t. $NM-[N,M]$ is a local martingale. By assumption, we have $[N,M]=-(NM-[N,M]-NM)$ is a local martingale. Protter says, it is a uniformly integrable martingale by the Kunita-Watanabe inequality. But i don't get it?
 A: To answer your question, mention several facts.
1. Let $X$ be a local martingale. If $\mathsf{E}[X^\ast_\infty]<\infty$, then $X$ is uniformly integrable martingale.
(ibid. Th.I.51, p.38) 
2. Suppose $X,Y\in\mathcal{H}^2$, then(ibid. Th.II.25 and Cor., p.69-70. and take H=K=1)
\begin{gather}
\mathsf{E}[([X,Y])^\ast_\infty]\le\mathsf{E}\biggl[\int_0^\infty|d[X,Y]_s|\biggr]\le (\mathsf{E}[X,X]_\infty)^{1/2}(\mathsf{E}[Y,Y]_\infty)^{1/2}
=\|X\|_{\mathcal{H}^2}\|Y\|_{\mathcal{H}^2}<\infty.\tag{1}\\
\mathsf{E}[X^\ast_{\infty}Y^\ast_{\infty}]\le \{\mathsf{E}[(X^\ast_{\infty})^2]\}^{1/2}\{\mathsf{E}[(Y^\ast_{\infty})^2]\}^{1/2}
\le 4\|X\|_{\mathcal{H}^2}\|Y\|_{\mathcal{H}^2}<\infty. \tag{2}
\end{gather}
3. Let $X,Y\in\mathcal{H}^2$ be two martingale, then using integration by parts we have (ibid. Cor.II.2, p.68)
$$ XY=X_-\centerdot Y+Y_-\centerdot X+[X,Y].$$
and $XY$ is a local martingale iff $[X,Y]$ is a local martingale, 
since the stochastic integrals $X_-\centerdot Y,Y_-\centerdot X$ are local martingale.  
$\impliedby$ Since $[X,Y]$ is a local martingale, then $XY$ is a local martingale and uniformly integrable(by (2)). 
$\implies$ Since $XY$ is a local martingale, then $[X,Y]$ is a local martingale and uniformly integrable(by (1)).
