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I know that if, for example, $(X, Y)$ is jointly normal, then any linear combination of them is normal.
Now my question is, is a collection of distinct pairwise sums jointly normal? If not, what kind of conditions are required for them to be jointly normal?
Affine transformations of multivariate normals are again multivariate normal. In your case,
$$\begin{pmatrix}X_1-X_4 \\ X_2-X_3 \\X_4-X_3 \end{pmatrix} = \underbrace{\begin{pmatrix} 1 & 0 & 0 & -1 \\0 & 1& -1 & 0 \\ 0 & 0 & -1 & 1 \end{pmatrix}}_{=:M} \underbrace{\begin{pmatrix}X_1 \\ X_2 \\ X_3 \\X_4 \end{pmatrix}}_{=:X} $$ is normal with mean $M\vec{\mu}$ and variance $M\Sigma M^t$, where $\vec\mu$ denotes the mean and $\Sigma$ the covariance matrix of $X$.
