Theorem. Quadratic elements realize the orthogonal Lie algebra [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000R]
Theorem. Quadratic elements realize the orthogonal Lie algebra [meinrenken2013clifford, Section 2.2.10, Proposition 2.12, p. 40] [fcap-000R]
Meinrenken states this realization over a characteristic-zero field. TauCeti proves the extension below over any field of characteristic different from \(2\). Let \(K\) be such a field, let \(V\) be finite-dimensional, and let \(Q\) be nondegenerate. Write \(\mathfrak {so}(V,B_Q)\) for the Lie algebra of endomorphisms skew-adjoint for the polar form. Then commutation on generators gives a Lie equivalence \[\mathfrak {so}(V,B_Q)\simeq _{\mathrm {Lie}}\mathfrak {cl}^{(2)}(Q).\] If \(A\in \mathfrak {so}(V,B_Q)\) corresponds to \(z_A\in \mathfrak {cl}^{(2)}(Q)\), then \[[z_A,\iota (x)]=\iota (Ax)\qquad (x\in V).\] Combining this equivalence with Theorem [fcap-000Q] gives the basis-free chain \[\bigwedge _K^2V\simeq _{\mathrm {Lie}}\mathfrak {so}(V,B_Q) \simeq _{\mathrm {Lie}}\mathfrak {cl}^{(2)}(Q).\] Kostant identifies the same degree-two subspace with \(\operatorname {Lie}(\operatorname {Spin}(V))\) and the first equivalence with the differential of the double cover [kostant1997clifford, Section 2.4, Theorem 8, pp. 286--287].