Matrix lie rings that contains a one-dimentional lie algebra of semi-simple matrices

Abstract

Let kk be a field and ¯¯¯kk¯ an algebraic closure of kk. Suppose that kk
contains more than five elements if char k≠2k≠2. Let hh be a one-dimensional subalgebra of the Lie k−k−algebra sl2¯¯¯ksl2k¯ consisting of semi-simple matrices. In this paper, it is proved that if g is a subring of the Lie ring sl2¯¯¯ksl2k¯ containing h, then g is either solvable or there exists a quaternion algebra A over a subfield FF of ¯¯¯kk¯ such that F⊇kF⊇k and g is isomorphic to the Lie F−F−algebra of all elements in A that are skew-symmetric with respect to a symplectic type involution defined on A.