GAP distinguishs between parent algebras and subalgebras of parent More about Groups and Subgroups), so here it is only sketched.
Each subalgebra belongs to a unique parent algebra, the so-called parent of the subalgebra. A parent algebra is its own parent.
Parent algebras are constructed by Algebra and UnitalAlgebra,
subalgebras are constructed by Subalgebra and UnitalSubalgebra.
The parent of the first argument of Subalgebra will be the parent of the
constructed subalgebra.
Those algebra functions that take more than one algebra as argument
require that the arguments have a common parent. Take for instance
Centralizer. It takes two arguments, an algebra A and an algebra
B, where either A is a parent algebra, and B is a subalgebra of
this parent algebra, or A and B are subalgebras of a common parent
algebra P, and returns the centralizer of B in A. This is
represented as a subalgebra of the common parent of A and B.
Note that a subalgebra of a parent algebra need not be a proper
subalgebra.
An exception to this rule is again the set theoretic function
Intersection (see Intersection), which allows to intersect algebras
with different parents.
Whenever you have two subalgebras which have different parent algebras
but have a common superalgebra A you can use AsSubalgebra or
AsUnitalSubalgebra (see AsSubalgebra, AsUnitalSubalgebra) in order
to construct new subalgebras which have a common parent algebra A.
Algebras and Unital Algebras).
The following sections describe the functions related to this concept (see Algebra, UnitalAlgebra, IsAlgebra, IsUnitalAlgebra, AsAlgebra, AsUnitalAlgebra, Subalgebra, UnitalSubalgebra, AsSubalgebra, AsUnitalSubalgebra, and also IsParent, Parent).
GAP 3.4.4