The largest coefficient of the highest root and the second smallest exponent

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.