Maximum Likelihood Estimation: Multivariate Gaussian function. Matrix calculus

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I am reading a paper and trying to understand how the authors estimated the standard errors of a set of parameter estimates $[\delta\ \ \phi \ \ \Sigma]$. Below is the loglikelihood function (sorry I could not typeset): enter image description here

where,

enter image description here

Using the trace trick and the rules of differentiation I managed to derive the first order conditions (after reading this). The authors provided the final results, which helped to verify my results. But I have been unable to derive the second order partial derivatives matrix i.e. Hessian. The authors do not derive the Hessian. I want to estimate the standard errors of the parameter estimates using the Hessian.

Will anyone spare a few minutes to help?

If you know any other method for estimating the standard errors of the parameter estimates, do please explain or provide reference.