Order conditions for stochastic Runge-Kutta methods preserving quadratic invariants of Stratonovich SDEs

In this paper we prove that for a stochastic Runge-Kutta method, the conditions for preserving quadratic invariants work as simplifying assumptions. For such methods, the method coefficients only have to satisfy one condition for each unrooted tree. This is a generalization of the result obtained for deterministic Runge-Kutta methods by Sanz-Serna and Abia in 1991.