The squared root lengths and inner product are
$$
|\alpha_1|^2=1,\qquad |\alpha_2|^2=3,\qquad
\alpha_1\mathbin\cdot\alpha_2=-\frac32.
$$
With $A_{ij}=2\alpha_i\mathbin\cdot\alpha_j/|\alpha_j|^2$, the <Cartan matrix> is
$$
\boxed{A=\begin{pmatrix}2&-1\\-3&2\end{pmatrix}}.
$$
This is the Cartan matrix of the <G2 root system>, with $\alpha_1$ short and $\alpha_2$ long.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 302 / 1 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-302/1/i/solution