Summary: Suppose we are given a finite-dimensional vector space \(V\) equipped with an \(F\)-rational action of a linearly algebraic group \(G\), with \(F\) a characteristic zero field. We conjecture the following: to each vector \(v\in V(F)\) there corresponds a canonical \(G(F)\)-orbit of semisimple vectors of \(V\). In the case of the adjoint action, this orbit is the \(G(F)\)-orbit of the semisimple part of \(v\), so this conjecture can be considered a generalization of the Jordan decomposition. We prove some cases of the conjecture.