Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 302 / 4 / c

The classification of rank-two root systems gives $A_1\times A_1$, $A_2$, $B_2$, and $G_2$. The first is reducible and corresponds to a semisimple but nonsimple algebra. Hence the complex simple rank-two algebras are $A_2$, $B_2=C_2$, and $G_2$, so $N=3$.