  It is well known that the presence of multiple constraints of non-Abelian
relativisitic Chern--Simons--Higgs vortex equations makes it difficult to
develop an existence theory when the underlying Cartan matrix $K$ of the
equations is that of a general simple Lie algebra and the strongest result in
the literature so far is when the Cartan subalgebra is of dimension 2. In this
paper we overcome this difficulty by implicitly resolving the multiple
constraints using a degree-theorem argument, utilizing a key positivity
property of the inverse of the Cartan matrix deduced in an earlier work of
Lusztig and Tits, which enables a process that converts the equality
constraints to inequality constraints in the variational formalism. Thus this
work establishes a general existence theorem which settles a long-standing open
problem in the field regarding the general solvability of the equations.
