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This is the G2 root system, with $\alpha$ long and $\beta$ short. Its positive roots areand the full root set includes their negatives. Thus $\dim \mathfrak{g}=12+2=14$. Finallyand simple roots have obtuse angle, so $\theta =5\pi /6$.

Choose a short simple root ${\alpha }_1$ and a long simple root ${\alpha }_2$, with $|{\alpha }_2{|}^2=3|{\alpha }_1{|}^2$ and angle $150^{\circ }$. The six positive roots of the G2 root system areand their negatives complete the two concentric hexagons of short and long roots. The fundamental weights areso ${\omega }_1$ is itself a short root and ${\omega }_2$ is the highest root.

The seven-dimensional representation $L_{{\omega }_1}$ has weight seteach with weight multiplicity one. Their positive heights are $1,2,3$, so the q-character of a highest-weight representation isThis is one ${\mathfrak{sl}}_2$ weight string, hence

The representation $L_{{\omega }_2}$ is the fourteen-dimensional Adjoint representation. Its nonzero weights are the twelve roots, and its zero-weight space is the two-dimensional Cartan subalgebra. The positive root heights are $1,1,2,3,4,5$, soSplitting this into ordinary ${\mathfrak{sl}}_2$ strings gives

Use ${\mathrm{ch}}_{q}V={\mathrm{tr}}_V{q}^{2{\rho }^{\vee }}$. The positive roots of a root system of the G2 root system have simple-root heights $1,1,2,3,4,5$. Each root space has dimension one, and the zero weight space has dimension two. ConsequentlyBy the classification of finite-dimensional sl2 representations, restriction to the Principal sl2 subalgebra is $L_{10}\oplus L_2$, of dimensions eleven and three. If the grading uses ${\rho }^{\vee }$ instead, divide every exponent by two.

If two simple roots have angle $5\pi /6$, normalize them to unit vectors $u,v$ with $(u,v)=-\sqrt{3}/2$. For any other unit simple root $w$, set $r=(u,w)\le 0$, $s=(v,w)\le 0$. Its three-root Gram matrix has determinant $1/4-r^2-s^2-\sqrt{3}rs$. Any nonzero $r$ or $s$ has absolute value at least $1/2$, contradicting positive definiteness. Thus all other simple roots are orthogonal to $u,v$. An irreducible root system containing this triple bond must be the rank-two G2 root system.