Solution
= Solution
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 $\theta=5\pi/6$.