Theorem. Exterior bivectors are the quadratic Clifford Lie algebra [meinrenken2013clifford, Section 2.2.10, Theorem 2.1, pp. 39--40] [fcap-000Q]
Theorem. Exterior bivectors are the quadratic Clifford Lie algebra [meinrenken2013clifford, Section 2.2.10, Theorem 2.1, pp. 39--40] [fcap-000Q]
Over the commutative ring and module of Definition [fcap-000O], let \(\mathfrak {cl}^{(2)}(Q)\) be the submodule of \(\mathcal {C}\kern -2pt\ell (Q)\) spanned by the elements \(\beta _Q(u,v)\). This is TauCeti's module-general extension of Meinrenken's characteristic-zero vector-space construction. The commutator formula of Lemma [fcap-000P], together with the Jacobi identity, shows that this submodule is closed under commutators. Transporting its bracket across \(\beta _Q\) gives a Lie algebra structure on \(\bigwedge _R^2M\) and a Lie equivalence \[\bigwedge _R^2M\simeq _{\mathrm {Lie}}\mathfrak {cl}^{(2)}(Q).\] On a decomposable bivector the equivalence is exactly \[u\wedge v\longmapsto \frac 12(uv-vu).\] No choice of basis enters this identification.