Construction of Multivector Inverse for Clifford Algebras Over $$2 m+1$$2 m+1-Di mensional Vector Spaces fro m Multivector Inverse for Clifford Algebras Over 2 m-Di mensional Vector Spaces

Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space $$\mathbb {R}^{p',q'}$$ , $$n'=p'+q'=2m$$ , we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over $$\mathbb {R}^{p,q}$$ , $$n=p+q=p'+q'+1=2m+1$$ . 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.