We consider the M(2, 3) minimal Liouville gravity, whose state space in the gravity sector is realized as irreducible modules of the Virasoro algebra. We present a recursive construction for BRST cohomology classes based on using an explicit form of singular vectors in irreducible modules of the Virasoro algebra. We find a certain algebra acting on the BRST cohomology space and use this algebra to find the operator algebra of physical states.