I know that marginal normals do not imply joint normal, as some examples in here gives. However, but I don't know of any theorems that talk about conditions under which marginal normals can imply normals. The solution in here is not satisfying (they only give a necessary condition).
In summary, I have two questions
Suppose $X_1$ and $X_2$ are random vectors, each follows a (different) normal distribution. What are the conditions for $(X_1, X_2)^T$ to be normal? Does this only require a positive definite covariance matrix?
Suppose $X_1, X_2, ..., X_n$ are univariate RVs and the conditionals $X_i|X_{-i}$ are normals for all $i$ where $X_{-i}$ denotes the vector $X = (X_1, ..., X_n)^T$ without $X_i$. Under which conditions will $X$ be normal?
