  The theory of abstract kernels in non-trivial extensions for many kinds of
algebraical objects, such as groups, rings and graded rings, associative
algebras, Lie algebras, restricted Lie algebras, DG-algebras and DG-Lie
algebras, has been widely studied since 1940's. Gerhard Hochschild firstly
treats associative algebra as an generic type in the series of kernel problems.
He proves the theorem of constructing kernel by presenting many tedious
relations that may lost the readers today. In this paper, we shall illustrate
the formulation and recast it for Lie algebra(-oid) kernels. We also prove the
independence of 3-cocycle in the case of associative algebra. Finally, we use
the universal enveloping algebra of Lie algebra to reduce the difficulty of a
direct construction for the derivation algebras.
