Let $\Phi \subset E$ be an irreducible reduced root system in a finite-dimensional real Euclidean [vector space](/page/Vector%20Space) $E$. Then $\Phi$ is isomorphic to exactly one root system of the following types: $A_n$ for $n \geq 1$, $B_n$ for $n \geq 2$, $C_n$ for $n \geq 3$, $D_n$ for $n \geq 4$, $E_6$, $E_7$, $E_8$, $F_4$, or $G_2$.