  Inspired by Fock-Goncharov's amalgamation procedure
\cite{Fock-Goncharov-2006}, Shen-Weng introduced string diagrams in
\cite{Shen-Weng-2021}, which are very useful to describe many interesting
skew-symmetrizable matrices closely related with Lie theory. In this paper, we
prove that the skew-symmetrizable matrices from string diagrams are in the
smallest class $\mathcal P^\prime$ of skew-symmetrizable matrices containing
the $1\times 1$ zero matrix and closed under mutations and source-sink
extensions. This result applies to the exchange matrices of cluster algebras
from double Bruhat cells, unipotent cells, double Bott-Samelson cells and so
on.
  Our main result can be used to explain why many skew-symmetrizable matrices
from Lie theory have reddening sequences. It can be also used to prove some
interesting results regarding non-degenerate potentials on many quivers from
Lie theory.
