Simple, locally finite dimensional Lie algebras in positive characteristic

We prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic p which give sufficient conditions for the algebras to be of the form [R(−),R(−)]/(Z(R)∩[R(−),R(−)]) or [K(R,⁎),K(R,⁎)] for a simple, locally finite dimensional associative algebra R with involution ⁎. The first proves that a condition we introduce, known as locally nondegenerate, along with the existence of an ad-nilpotent element suffice. The second proves that an ad-integrable Lie algebra is of this type if the characteristic of the ground field is sufficiently large. Lastly we construct a simple, locally finite dimensional associative algebra R with involution ⁎ such that K(R,⁎)≠[K(R,⁎),K(R,⁎)] to demonstrate the necessity of considering the commutator in the first two theorems.