  In the traditional Katz-Vafa method, matter representations are determined by
decomposing the adjoint representation of a parent simple Lie algebra
$\mathfrak{m}$ as the direct sum of irreducible representations of a semisimple
subalgebra $\mathfrak{g}$. The Katz-Vafa method becomes ambiguous as soon as
$\mathfrak{m}$ contains several subalgebras isomorphic to $\mathfrak{g}$ but
giving different decompositions of the adjoint representation. We propose a
selection rule that characterizes the matter representations observed in
generic constructions in F-theory and M-theory: the matter representations in
generic F-theory compactifications correspond to linear equivalence classes of
subalgebras $\mathfrak{g}\subset \mathfrak{m}$ with Dynkin index one along each
simple components of $\mathfrak{g}$. This simple yet elegant selection rule
allows us to apply the Katz-Vafa method to a much large class of models. We
illustrate on numerous examples how this proposal streamlines the derivation of
matter representations in F-theory and resolves previously ambiguous cases.
