LCS-Baer correspondence between inner derivations and inner automorphisms for 2-Engel Lie ring

Statement
Suppose $$L$$ is a 2-Engel Lie ring. Then, the following are true:


 * 1) $$L$$ is a Lie ring in which every inner derivation is exponentiable, i.e., every inner derivation of $$L$$ is an exponentiable derivation.
 * 2) The quotient $$L/Z(L)$$, which is the Lie ring of inner derivations of $$L$$, is a Lie ring of nilpotency class two. In fact, this quotient must be either an abelian Lie ring or a Lie ring of nilpotency class two whose center has exponent three. In particular, it is a LCS-Baer Lie ring and hence a LCS-Lazard Lie ring.
 * 3) The exponential defines a bijection from the Lie ring $$L/Z(L)$$ of inner derivations of $$L$$ to the group of inner automorphisms of $$L$$.
 * 4) The bijection defined by the exponential gives a LCS-Baer correspondence, the class two version of the LCS-Lazard correspondence.