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

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