A Grassmann algebra with three generators:

Compute the canonical form of an element of the algebra:

A Grassmann algebra with four generators:

Compute the Gröbner basis of a left ideal in the algebra:

Specify a nondefault multiplication operation:

Collect terms involving the same powers of :

Define the exterior algebra in two variables:

Confirm the basic relations , and :

Any product with three or more generators in it will contain a repeated generator and thus be zero:

The terms , , and are linearly independent—NonCommutativeExpand finds no relations—forming a basis for the algebra:

Compute the product of two general linear combinations of the generators:

The algebra ext can be represented using TensorWedge with the generators and corresponding to 2-vectors:

Confirm the basic relations; in this concrete realization, the product is represented by a zero matrix rather than a zero scalar:

Compute the product of two general vectors; TensorReduce gives a result corresponding to the one found abstractly:

Repeat the computation with two explicit vectors; the result is an antisymmetric rank-2 array (matrix):

The entries of the array are , where is the determinant of the coefficients and is Signature[i,j]:

GrassmannAlgebra gives a NonCommutativeAlgebra with Grassmann algebra structure: