  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 $[R^{(-)}, R^{(-)}] /
(Z(R) \cap [R^{(-)}, R^{(-)}])$ 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 a
uniformly 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 $K(R,
*) \ne [K(R, *), K(R, *)]$ to demonstrate the necessity of considering the
commutator in the first two theorems.
