In a recent post I commented on the connection between the multivariate normal distribution and marginal distributions that are normal. Specifically, the latter do not necessarily imply the former.
So, let's think about this in terms of testing for normality.
Suppose that we have several variables which we think may have a joint distribution that's normal. We could test each of the variables for normality, separately, perhaps using the Jarque-Bera LM test. If the null hypothesis of normality was rejected for one or more of the variables, this could be taken as evidence against multivariate normality. However, if normality couldn't be rejected for any of the variables, this wouldn't tell us anything about their joint distribution.
What we need is a test for multivariate normality itself. Let's see what's available.