Let $$f_{X,Y}(x,y)=\frac{\sqrt{3}}{4\pi}\exp\left(-\frac{x^{2}-xy+y^{2}}{2}\right)$$ and find $a,b,c,d$ for which $U=aX+bY, V=cX+dY$ are independent.
I know $X$ and $Y$ are correlated, with variance of $X, Y$ is $3/4$, and $Cov(X,Y)=2/3$. So, my claim is that $U$ and $V$ are bivariate normal and if their covariance is zero, then $U$ and $V$ are independent.
But, I have difficulty in showing $U$ and $V$ are bivariate normal.
Hint: In general, if $Z\in\mathbb{R}^n$ is $Z\sim N(\mu,V)$, then for any nonstochastic $M$ that has $n$ columns, we have $MZ\sim N(M\mu,MVM')$. In this case, $$ Z=\begin{pmatrix} X,Y \end{pmatrix},\quad \mu=\begin{pmatrix}0\\0\end{pmatrix},\quad \color{red}{V=\begin{pmatrix} \frac{4}{3}&\frac{2}{3}\\ \frac{2}{3}&\frac{4}{3} \end{pmatrix}},\quad M=\begin{pmatrix}a&b\\ c&d\end{pmatrix}. $$ You're to find $M$ such that $MVM'$ is diagonal, which, after simplifications, entails $$ \color{red}{\boxed{2 a c + b c + a d + 2 b d=0.}} $$ Any $a,b,c,d$ satisfying the above will make $U,V$ independent.