The vectors of a root-system were originally called "roots" because they are the zeros of a characteristic polynomial that comes from the connection of root-systems to classifying semisimple Lie algbras. From an answer by José Carlos Santos,

It comes from the roots of the characteristic polynomial of an endomorphism. If $\mathfrak g$ is a complex semisimple Lie algbra, $\mathfrak h$ is a Cartan subalgebra and $\alpha\in\mathfrak{h}^*$, then $\alpha$ is a root if, for every $H\in\mathfrak h$, $\alpha(H)$ is an eigenvalue of the endomorphism of $\mathfrak g$ defined by $X\mapsto[H,X]$.

Is there a way to get at this same polynomial just from the definitions of a root-system?