An indefinite stochastic Riccati equation is a matrix-valued, highly nonlinear backward stochastic differential equation together with an algebraic, matrix positive definiteness constraint. We introduce a new approach to solve a class of such equations (including the existence of solutions) driven by one-dimensional Brownian motion. The idea is to replace the original equation by a system of backward stochastic differential equations (without involving any algebraic constraint) whose existence of solutions automatically enforces the original algebraic constraint to be satisfied.