Virasoro closure from BRST nilpotence ($Q^2=0⟹[L_m,L_n]=(m-n)L_{m+n}$)

If $L_m=\{b_m,Q\}$, nilpotence gives $[L_m,Q]=0$. The graded Jacobi identity and $[b_m,L_n]=(m-n)b_{m+n}$ then imply the displayed algebra. Thus a quantum central extension in the total generators obstructs BRST nilpotence.