Calculate $\mathbb{E}(X-Y\mid 2X+Y).$ if $X\sim N(0,a)$ and $Y\sim N(0,b)$ 
Question: Given that $X$ and $Y$ are two random variables  satisfying $X\sim N(0,a)$ and $Y\sim N(0,b)$ for some $a,b>0$. Assume that $X$ and $Y$ have correlation $\rho.$ 
  Calculate 
  $$\mathbb{E}(X-Y \mid 2X+Y).$$

I tried to use the fact that if $A$ and $B$ are independent, then $\mathbb{E}(A\mid B) = \mathbb{E}(A)$ and uncorrelated implies independence in jointly normal distribution. 
So, I attempted to express $X-Y$ as a linear combination of $2X+Y$ and $Z$ where $\operatorname{Cov}(2X+Y,Z) = 0.$
But I am not able to do so.
Any hint is appreciated.

The joint distribution of $(Z_1,Z_2)\equiv(X-Y,2X+Y)$ is $\mathcal{N}(0,\Sigma)$, where
$$
\Sigma=\begin{bmatrix}
a+b-2\rho\sqrt{ab} & 2a-b-\rho\sqrt{ab} \\
2a-b-\rho\sqrt{ab} & 4a+b+4\rho\sqrt{ab}
\end{bmatrix}.
$$
Then the conditional distribution of $Z_1$ given $Z_2$ is 
$$
Z_1\mid Z_2=z\sim \mathcal{N}(\Sigma_{12}\Sigma_{22}^{-1}z,\,\Sigma_{11}-\Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}).
$$
