It is well-known that the following is one of the equivalent definitions of "jointly normal":
Definition 1. Let $(X_1,X_2)$ be a random vector. Then $X_1$ and $X_2$ are said to be jointly normal if and only if there exist real numbers $\mu_1, \mu_2, \alpha_1, \ldots, \alpha_k, \beta_1,\ldots, \beta_k$ and independent random variables $Z_1,Z_2, \ldots , Z_k$ (for some $k \geq 1$), each of which has a standard normal distribution, such that
$$ \begin{align*} X & = \sum_{i=1}^k \alpha_i Z_i + \mu_1 \\ Y & = \sum_{i=1}^k \beta_i Z_i + \mu_2. \end{align*} $$
I think (not entirely sure, correct me if I'm wrong) the following statement ("Definition 2") is equivalent to Definition 1:
Definition 2. Let $(X_1,X_2)$ be a random vector. Then $X_1$ and $X_2$ are said to be jointly normal if and only if there exist real numbers $ \alpha_1, \ldots, \alpha_k, \beta_1,\ldots, \beta_k$ and independent random variables $Z_1,Z_2, \ldots , Z_k$ (for some $k \geq 1$), each of which has a normal distribution, such that
$$ \begin{align*} X & = \sum_{i=1}^k \alpha_i Z_i \\ Y & = \sum_{i=1}^k \beta_i Z_i. \end{align*} $$
Notice that, in contrast to Definition 1, in Definition 2 the rvs $Z_i$ need not necessarily have the standard normal distribution (any normal is fine).
I want to show the equivalence of the two definitions given above. It is obvious that Definition 2 implies Definition 1. I have not been able to show that Definition 1 implies Definition 2; I do have some partial results, namely under the extra assumptions that $k=2$ and $\alpha_1 \beta_2 - \alpha_2 \beta_1 \neq 0$ I managed to prove it. Of course the general idea is to "absorb" the $\mu$s inside the rvs, but unfortunately I haven't been able to do it. Can someone point out a proof or sketch? Thanks a lot. Alternatively, if someone knows a reference where this is shown, that would also be great!
We have equivalence of the two definitions provided that any of the $Z_i$ in Definition 2 is allowed to be a degenerate normal variable with standard deviation zero, i.e., a constant. Notice that under Definition 1 it's possible to achieve a jointly normal distribution $(X,Y)$ where both $X$ and $Y$ are nonzero constants. But this is never possible if every $Z_i$ in Definition 2 is required to be nondegenerate normal.
So given that we can take $Z_{k+1}=1$ in Definition 2, the proof that Definition 1 implies Definition 2 is trivial.