Example. The degree-two symbol [chevalley1954algebraic, II.1.6, pp. 41--42] [fcap-000K]
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filtrationAssociatedGradedEquivExterior_apply_of_succ
L∃∀N

For \(u,v\in M\), write the Clifford product as \[\iota (u)\iota (v) =\frac 12\bigl (\iota (u)\iota (v)-\iota (v)\iota (u)\bigr ) +\frac 12B_Q(u,v)1.\] The second summand belongs to \(F_0\). Hence the degree-two class of \(\iota (u)\iota (v)\) is \(u\wedge v\). This is the two-generator instance of Chevalley's leading-symbol calculation cited in the title. The surviving half-commutator is the Clifford bivector studied next; the scalar polar term is invisible to the leading symbol.

Example. The degree-two symbol [chevalley1954algebraic, II.1.6, pp. 41--42] [fcap-000K]
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filtrationAssociatedGradedEquivExterior_apply_of_succ
L∃∀N

Reference. The algebraic theory of spinors [chevalley1954algebraic]

The exterior shadow of a Clifford algebra [fcap-000F]

Definition. Word-length filtration [chevalley1954algebraic, II.1.2 and II.1.6, pp. 40--42] [fcap-000G]

filtration filtration_mul
L∃∀N

Lemma. The leading symbol is exterior [chevalley1954algebraic, II.1.6, pp. 41--42] [fcap-000H]

changeForm_prod_map_ι_sub_prod_map_ι_mem_filtration filtrationLeadingTerm_apply_ιMulti
L∃∀N

Proof.

Theorem. The graded pieces are exterior powers [chevalley1954algebraic, II.1.6, pp. 41--42] [fcap-000I]

filtrationGradedPieceEquivExteriorPower filtrationLeadingTerm_eq_filtrationGradedEquiv_symm
L∃∀N

Theorem. PBW equivalence with the exterior algebra [meinrenken2013clifford, Proposition 2.6, pp. 33--34] [fcap-000J]

filtrationAssociatedGradedEquivExterior
L∃∀N

Example. The degree-two symbol [chevalley1954algebraic, II.1.6, pp. 41--42] [fcap-000K]
✍️source
filtrationAssociatedGradedEquivExterior_apply_of_succ
L∃∀N

Remark. Basis proof and quotient proof [fcap-000L]