It is proved in this paper that covariance hypotheses which are linear in both the covariance and the inverse covariance are products of models each of which consists of either (i) independent identically distributed random vectors which have a covariance with a real, complex or quaternion structure or (ii) independent identically distributed random vectors with a parametrization of the covariance which is given by means of the Clifford algebra. The models (i) are well known. For models (ii) we have found, under the assumption that the distribution is normal, the exact distributions of the maximum likelihood estimates and the likelihood ratio test statistics.
"Covariance Hypotheses Which are Linear in Both the Covariance and the Inverse Covariance." Ann. Statist. 16 (1) 302 - 322, March, 1988. https://doi.org/10.1214/aos/1176350707