Let $X,Y \in \lambda^2_{loc}$ and let $M_t=\int_0^tX_sdB_s^i $ and $M_t=\int_0^tY_sdB_s^j$, with $i\neq j$.
With $B_s^i$ and $B_s^j$ I indicate the one dimensional B.M. of a $d$ dimensional B.M. $B=(B_t^1...B_t^d)$
How could I prove that $\langle M,N \rangle_t=0$?? I think I have to refer to the definition of the stochastic integral with simple process but I'm not sure.
Since $B$ is a $d$-dimensional Brownian motion, $B_s^i, B_s^j$ are independent $1$-dimensional Brownian motions. This implies that $[B_s^i, B_s^j]_t = 0$ (see the section on quadratic variation of B.M. here).
The result now follows immediately since by standard properties of the Ito integral, we have that $[M,N]_t = \int_0^t M_s N_s d [ B^i, B^j]_s = 0$.