Construction of multivector inverse for Clifford algebras over 2m+1-dimensional vector spaces from multivector inverse for Clifford algebras over 2m-dimensional vector spaces

Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space Rp′,q′, n ′ = p ′ + q ′ = 2 m, we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over R p,q , n= p+ q= p ′ + q ′ + 1 = 2 m+ 1. Explicit examples are provided for dimensions n ′ = 2 , 4 , 6 , and the resulting inverses for n= n ′ + 1 = 3 , 5 , 7. The general result for n= 7 appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(p, q), n= p+ q= 7 , only involving a single addition of multivector products in forming the determinant.