Free algebras of the Hilbert modular forms

The talk is based on the joint work with E. Stuken (HSE). Let $K=\mathbb{Q}\sqrt{d}$ (for $d$ a square-free integer) be a real quadratic field, $A$ is the ring of all algebraic integers in $K$. Consider the Hilbert modular group $ \Gamma_{d}={\mathrm{PSL}}(2,A)$ acting as a discrete group of automorphisms on the product $H \times H$ of two upper half planes. Let $\tau$ be the trasposition of the half planes and $\widehat\Gamma$ be the group generated by $\Gamma_{d}$ and $\tau$.
Denote by $A(\widehat\Gamma)$ the algebra of $\widehat\Gamma$ -automorphic forms on $H \times H$. The main goal of the report is the following
Theorem. If the algebra $A(\widehat\Gamma)$ is free then $d \in (2,3,5,6,13,21)$.
I would like to discuss this result as a part of more general problem: find the all discrete lattices acting on homogenous hermitian Cartan domains of type four with free algebras of automorphic forms.
Remarkable contributions to the subject was recently made by E. Vinberg.