Irreducible linear groups of degree four over a quaternion division algebra that contain a subgroup diag(T3(K,Φ0),1)

Let F be a field of characteristic ≠2, D a quaternion division algebra over F, K a subfield of D, σ an involutory automorphism of K, Φ0 a non-degenerate σ-skew-Hermitian form in three variables over K whose Witt index is 1. Suppose that K contains a subfield K0 of F such that each element of K0 is invariable under σ and F is algebraic over K0. We study the irreducible subgroups of GL4(D) that contain the subgroup diag(T3(K,Φ0),1).