Distribution of $(X_1,X_2)$ if $X_1\pm X_2$ are two independent $N(1,4)$
Let $Y = (Y_1,Y_2)$ be two i.i.d. $N(1,4)$ variables. Then $Y$ is a multivariate normal distribution, and so any affine transformation of $Y$ is a multivariate normal distribution. In particular, if we define $X_1 = \frac{Y_1 + Y_2}{2}$ and $X_2 = \frac{Y_1 - Y_2}{2}$ then $X = (X_1,X_2)$ is a multivariate normal distribution which satisfies $X_1 + X_2 = Y_1$ and $X_1 - X_2 = Y_2$. In other words, it is the random variable you're interested in.
It is routine to calculate
$$
(X_1,X_2) \sim N\left(\begin{pmatrix} 1 &amp; 0 \end{pmatrix}, \begin{pmatrix} 2 &amp; 0 \\ 0 &amp; 2 \end{pmatrix}\right),
$$
that is, $X_1 \sim N(1,2)$ and $X_2 \sim N(0,2)$, and $X_1,X_2$ are independent.