Crossvariation of two independent B.M.

Question

Let $B_1$ and $B_2$ be two independent B.M. We defined the Crossvariation of $X$ and $Y$ (if they quadratic Variation of both of them exist) as $ \langle X,Y \rangle_t = \frac{1}{4} ( \langle X+Y \rangle_t + \langle X-Y \rangle_t ) $ Hence i started to calculate $\langle X+Y \rangle_t$. This is $$ \langle X+Y \rangle_t = QV_{[0,t]}(B_1+B_2)= \sum_{j=1}^{len(\pi^{k})}((B_1+B_2)(\pi_j^{k})-(B_1+B_2)(\pi_{j-1}^{k}))^2= \sum_{j=1}^{len(\pi^{k})}(B_1(\pi_j^{k})-B_1(\pi_{j-1}^{k})+ B_2(\pi_j^{k})-B_2(\pi_{j-1}^{k}))^2= \sum_{j=1}^{len(\pi^{k})} ((B_1(\pi_j^{k})-B_1(\pi_{j-1}^{k}))^2+ \sum_{j=1}^{len(\pi^{k})} ((B_2(\pi_j^{k})-B_2(\pi_{j-1}^{k}))^2+2\sum_{j=1}^{len(\pi^{k})}(B_1(\pi_j^{k})-B_1(\pi_{j-1}^{k})(B_2(\pi_j^{k})-B_2(\pi_{j-1}^{k})) = t+t+ 2\sum_{j=1}^{len(\pi^{k})}(B_1(\pi_j^{k})-B_1(\pi_{j-1}^{k})(B_2(\pi_j^{k})-B_2(\pi_{j-1}^{k})) $$ Now i dont know how to calculate the last Term. The same is for $\langle B_1-B_2 \rangle_t$. Maybe someone can help me, thanks alot!