Why is a root system called a "root" system?

Root systems plays an important role in, among other things, classifying semisimple Lie Algebras. Their name suggest that they have something to do with "roots" of a polynomial. Are they the roots of some polynomial? Where does the name "root system" come from?

• roots of a characteristic poly of $t$ in a torus acting on the lie algebra- see almost exactly 5 minutes on, from approx the 15:00 mark in youtube.com/… of Gross's lecture Jan 26, 2018 at 14:29
• For the corresponding question regarding weights, cf. mathoverflow.net/q/154933/27465 Feb 2, 2018 at 6:37

It comes from the roots of the characteristic polynomial of an endomorphism. If $$\mathfrak g$$ is a complex semisimple Lie algebra, $$\mathfrak h$$ 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 $$X\mapsto[H,X]$$.
• ...except that, usually by fiat, $0$ is not a root. Jan 26, 2018 at 14:50