This is the <G2 root system>, with $\alpha$ long and $\beta$ short. Its positive roots are
$$
\beta,\ \alpha,\ \alpha+\beta,\ \alpha+2\beta,\ \alpha+3\beta,\ 2\alpha+3\beta,
$$
and the full root set includes their negatives. Thus $\dim\mathfrak g=12+2=14$. Finally
$$
4\cos^2\theta=A_{12}A_{21}=3,
$$
and simple roots have obtuse angle, so $\theta=5\pi/6$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-302/4/d/solution