Algebra group structures for dihedral group:D8

The group dihedral group:D8 has at least one (and probably only one?) algebra group structure over field:F2. It does not have any algebra group structure over any other fields.

Multiplication table (structure constants)
The algebra is a three-dimensional algebra. We can describe it by means of the following multiplication table in terms of structure constants $$u,v,w$$. The multiplication table is as follows:

Verification of properties

 * The algebra is associative: All products of length three or more are zero.
 * The algebra is nilpotent: All products of length three or more are zero.
 * The algebra group is isomorphic to dihedral group:D8: $$1 + w$$ is the central element of order 2, $$1 + u + v$$ is the order four generator of the cyclic maximal subgroup. $$1 + u$$ and $$1 + v$$ are reflections outside this subgroup.

Description as subalgebra of niltriangular matrix Lie algebra
The algebra is the whole of niltriangular matrix Lie algebra:NT(3,2), so dihedral group:D8 is isomorphic to $$UT(3,2)$$. The explicit description is:

$$u = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\\end{pmatrix}, \qquad v = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\\end{pmatrix}, \qquad w = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\\end{pmatrix}$$