A usual procedure in mathematics is the definition of new operations for given objects; here are a few typical examples. The Lie bracket defines an interesting new multiplicative structure on a given (associative) algebra. Forming a group ring can be viewed as defining a new addition for the elements of the given group, and extending the multiplication to sums of group elements in a natural way. Forming the exterior algebra of a given vector space can be viewed as defining a new multiplication for the vectors in a natural way.
GAP does not support such a procedure.
The main reason for this is that in GAP, the multiplication in a group,
a ring etc. is always written as *,
and the addition in a vector space, a ring etc. is always written as +.
Therefore it is not possible to define the Lie bracket as a
``second multiplication'' for the elements of a given algebra;
in fact, the multiplication in Lie algebras in GAP is denoted by *.
Analogously, constructing the group ring as sketched above is impossible
if an addition is already defined for the elements;
note the difference between the usual addition of matrices and the
addition in the group ring of a matrix group!
(See Chapter Magma Rings in the Reference Manual for an example.)
Similarly, there is already a multiplication defined for row vectors
(yielding the standard scalar product), hence these vectors cannot be
regarded as elements of the exterior algebra of the space.
In situations such as the ones mentioned above,
GAP's way to deal with the structures in question is the following.
Instead of defining new operations for the given objects,
new objects are created to which the given arithmetic operations
* and + are then made applicable.
With this construction, matrix Lie algebras consist of matrices that are
different from the matrices with associative multiplication;
technically, the type of a matrix determines how it is multiplied with
other matrices (see IsMatrix in the Reference Manual).
A matrix with the Lie bracket as its multiplication can be created with
the function LieObject
from a matrix with the usual associative multiplication.
Group rings (more general: magma rings,
see Chapter Magma Rings in the Reference Manual)
can be constructed with FreeMagmaRing from a coefficient ring
and a group.
The elements of the group are not contained in such a group ring,
one has to use an embedding map for creating a group ring element that
corresponds to a given group element.
It should be noted that the GAP approach to the construction of
Lie algebras from associative algebras is generic in the sense
that all objects in the filter IsLieObject use the same methods
for their addition, multiplication etc.,
by delegating to the ``underlying'' objects of the associative algebra,
no matter what these objects actually are.
Analogously, also the construction of group rings is generic.
GAP 4 manual