Show the the sums of independent random variables are independent using covariance

Question

Assume $X_1, X_2, X_3$ are independent random variables from a normal distribution with unknown mean $\mu$ and variance $\sigma^2$. Let $W_1=X_1-X_2$ and $W_2=X_1+X_2-2X_3$. I need to show that $W_1$ and $W_2$ are independent. Can I show this using the fact that if $Cov(W_1,W_2)=0$ then they are independent. My wokrings so far: $$\begin{array}CCov(W_1,W_2)=Cov(X_1-X_2,X_1+X_2-2X_3)\\ Cov(W_1,W_2)=\bigr[Cov(X_1,X_1)+Cov(X_1,X_2)+Cov(X_1,-2X_3)+Cov(-X_2,X_1)+Cov(-X_2,X_2)+Cov(-X_2,-2X_3)\bigr]\\Cov(W_1,W_2)= 2Cov(X_2,X_3)-2Cov(X_1,X_3)\\Cov(W_1,W_2)=E(X_2X_3)-E(X_1X_3)\end{array}$$ I'm not sure how to show that $E(X_2X_3)=E(X_1X_3)$

Answer 1

By independence we have $$ E(X_2 X_3) = E(X_2)E(X_3) = \mu^2 = E(X_1)E(X_3) = E(X_1 X_3)$$