Rank-two subsystems of a C3 root system ($A_2\text{ and }{C}_2$)

In the realization $\Phi (C_3)=\{\pm 2e_i,\pm e_i\pm e_j:i<j\}$, 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 $\Vert l{\Vert }^2=2\Vert s{\Vert }^2$ and $s\cdot l=-\Vert s{\Vert }^2$, 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.