  There are many different ways that the exponents of Weyl groups of
irreducible root systems have been defined and put into practice. One of the
most classical and algebraic definitions of the exponents is related to the
eigenvalues of Coxeter elements. While the coefficients of the height root when
expressed as a linear combination of simple roots are combinatorial objects in
nature, there are several results asserting relations between these exponents
and coefficients. This study was conducted to give a uniform and fairly
elementary proof of the fact that the second smallest exponent of the Weyl
group is one or two plus the largest coefficient of the highest root of the
root system depending upon a simple condition on the root lengths. As a
consequence, we obtain a necessary and sufficient condition for a root system
to be of type $G_2$ in terms of these numbers.
