= Rank-two subsystems of a C3 root system
{title2=$A_2\text{ and }C_2$}
In the realization $\Phi(C_3)=\{\pm2e_i,\ \pm e_i\pm e_j:i<j\}$, each coordinate two-plane cuts out a <C2 root system>. Each plane $\pm x_1\pm x_2\pm x_3=0$ cuts out an <A2 root system>; there are four such planes after identifying opposite normals. Thus the irreducible full rank-two <root subsystems> comprise three of type $C_2$ and four of type $A_2$. For an $A_2$ simple pair $p,q$, equal lengths and angle $120$ degrees give roots $\pm p,\pm q,\pm(p+q)$. For a mixed-length simple pair $s,l$ with $\|l\|^2=2\|s\|^2$ and $s\cdot l=-\|s\|^2$, the <B2 root system> has roots $\pm s,\pm l,\pm(s+l),\pm(2s+l)$. These lists also verify that the selected pairs are simple, rather than simply independent.
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