Geometric identities, invariant theory, and a theorem of Bricard (Q1336473)

A Peano space of step \(n\) is a pair \((V, [\cdot])\) where \(V\) is a vector space and \([\cdot]\) is a bracket of step \(n\) over \(V\). A Peano space of step \(n\) equipped with two operations of `joint' and `meet' is called the double algebra of step \(n\). The author proves an identity in the double algebra of a Peano space (Theorem 2.1). This yields a class of geometric identities valid in \(n\)- dimensional projective spaces. A theorem of Bricard in dimension two and one of Fontené in dimension three are special cases of the identity obtained in the paper.