  We use the numerical conformal bootstrap in two dimensions to search for
finite, closed sub-algebras of the operator product expansion (OPE), without
assuming unitarity. We find the minimal models as special cases, as well as
additional lines of solutions that can be understood in the Coulomb gas
formalism. All the solutions we find that contain the vacuum in the operator
algebra are cases where the external operators of the bootstrap equation are
degenerate operators, and we argue that this follows analytically from the
expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal
blocks. Our numerical analysis is a special case of the "Gliozzi" bootstrap
method, and provides a simpler setting in which to study technical challenges
with the method.
  In the supplementary material, we provide a Mathematica notebook that
automates the calculation of the crossing matrices and OPE coefficients for
degenerate operators using the formulae of Dotsenko and Fateev.
