Independence of linear combination of multivariate distribution

$X$ is multivariate normal distributed with $\mu = (2, 2)$ and $\Sigma = (1, 0 ; 0, 1)$ and I have 2 vectors $A = (1, 1)$ and $B = (1, -1)$. How can I than show that the linear combinations $AX$ and $BX$ are independent?

First of all independence follows from a lack of correlation when two random variables are jointly normal. Now it isn't hard to see that $$AX$$ and $$BX$$ are uncorrelated:
\begin{align} \text{Cov}(AX, BX) &= \text{Cov}(X_1 + X_2, X_1 - X_2) \\ &= \text{Var}(X_1) - \text{Var}(X_2) \\ &= 0 \end{align}
so we just need to show that $$AX$$ and $$BX$$ are jointly normal. To do that it's enough to show that all linear combinations of $$AX$$ and $$BX$$ are normally distributed. Can you finish the proof?