It seems to me that if a random vector $X$ is to have a multivariate normal distribution, then it is necessary and sufficient that $X$ is a vector of independent Gaussians -- is this correct?
In my understanding, a random vector $X$ of length $k$ is said to have a multivariate normal distribution if for any constant vector $a \in \mathbb{R}^k$, the random variable $Y = a^TX$ has a univariate normal distribution.
Then $e_i^TX$ must be a Gaussian for the elementary basis vectors $e_i$, implying that each $X_i$ must be a Gaussian.
In the other direction, any linear combination of two (or more, by induction) Gaussians should produce another Gaussian.