  A program of wide interest in modern conformal bootstrap studies is to
numerically solve general conformal field theories, based on a critical
assumption that the dynamics is encoded in the conformal four-point crossing
equations and positivity condition. In this letter we propose and verify a
novel algebraic property of the crossing equations which provides strong
restriction for this program. We show for various types of symmetries $\cal G$,
the crossing equations can be linearly converted into the $SO(N)$ vector
crossing equations associated with the $SO(N)\rightarrow \cal G$ branching
rules and the transformations satisfy positivity condition. The dynamics
constrained by the $\cal G$-symmetric crossing equations combined with
positivity condition degenerates to the $SO(N)$ symmetric cases, while the
non-$SO(N)$ symmetric theories are not directly solvable without introducing
the $SO(N)$ symmetry breaking assumptions on the spectrum.
