Exceptional Lie algebra G2

Description

The complex Lie algebra $\mathfrak{g}_2$ is the simple Lie algebra over $\C$ whose root system has type $G_2$. The root system $G_2$ lies in the plane $\R^2$ and has $12$ roots. The $6$ short roots have length $1$ and angles $0^\circ, 60^\circ, \dots, 300^\circ$ with the first axis. The $6$ long roots have length $\sqrt{3}$ and angles $30^\circ, 90^\circ, \dots, 330^\circ$. With a short simple root $\alpha_1$ and a long simple root $\alpha_2$ at an angle of $150^\circ$, the Cartan matrix with entries $\langle \alpha_i, \alpha_j \rangle$ is $$\begin{pmatrix} 2 & -1 \\ -3 & 2 \end{pmatrix}.$$ Check: $(\alpha_1, \alpha_2) = 1 \cdot \sqrt{3} \cdot \cos 150^\circ = -3/2$, so $\langle \alpha_1, \alpha_2 \rangle = 2(-3/2)/3 = -1$ and $\langle \alpha_2, \alpha_1 \rangle = 2(-3/2)/1 = -3$. The rank is $2$, and the dimension is the rank plus the number of roots, $2 + 12 = 14$.