# Internal semidirect product of Lie rings

From Groupprops

## Definition

Suppose is a Lie ring and and are Lie subrings of . We say that is an **internal semidirect product** of and , denoted , if it satisfies **all** the following conditions:

Conditions (3) and (4) basically say that the additive group of is the internal direct product of the additive groups of and .

This notion is equivalent to the notion of external semidirect product of Lie rings, via equivalence of internal and external semidirect product for Lie rings.