  We generalize a classical result of Andrianov on the decomposition of Hecke
polynomials. Let $\mathfrak{F}$ be a non-archimedean local fied. For every
connected reductive group $\mathbf{G}$, we give a criterion for when a
polynomial with coefficients in the spherical parahoric Hecke algebra of
$\mathbf{G}(\mathfrak{F})$ decomposes over a parabolic Hecke algebra associated
with a non-obtuse parabolic subgroup of $\mathbf{G}$. We classify the
non-obtuse parabolics. This then shows that our decomposition theorem covers
all the classical cases due to Andrianov and Gritsenko. We also obtain new
cases when the relative root system of $\mathbf{G}$ contains factors of types
$E_6$ or $E_7$.
