Consider a good integrator $X$ (semi-martingale) and the relative quadratic variation process indicated by: $Y_t:=[X,X]_t$.
Why is that:
$$[Y,Y]_t=0 \ \ \ \ \ and \ \ \ \ \ \ [X,Y]_t=0 \ \ ?$$
EDIT: I believe that for general good integrators the previous statement is not true. Indeed:
If $Y$ is adapted, cadlag, with paths of finite variation on compacts, then Y is a quadratic pure jump semimartingale. Therefore: $[Y,Y]_t=∑_{0<s<t}(ΔY_s)^2$.
Quadratic variation of a semi-martingale is non-decreasing and right-continuous. (Or at least there exists such a version of it). You know that a function is of bounded variation if and only if it is the difference of two non-decreasing functions. So $[X,X]_t$ is of bounded variation and right-continuous. Then its quadratic variation must be the sum of the jumps squared over the interval the Q.V. is computed.
If you impose further that the semi-martingale is continuous, then its Q.V. is continuous as well. The argument above gives in that case a Q.V. of $0$.