Suppose I have the vectors $\alpha, \beta \in \mathbb{R}^2$ with inner products $(\alpha, \alpha) = 1$ and $(\beta, \beta) = 2$, and the angle between $\alpha$ and $\beta$ is $\theta = \frac{3\pi}{4}$.
I want to show $R = \{ \alpha, \beta, -\alpha, -\beta, \alpha + \beta, -\alpha - \beta, 2\alpha + \beta, -2\alpha - \beta \}$ is a root system, that is it satisfies the following:
- $R$ spans $\mathbb{R}^2$
- $\forall x \in R$ and $r \in \mathbb{R}$, $rx \in R \iff r = 1$ or $r = -1$
- $\forall x \in R$, $\sigma_x(R) \subset R$
- $\forall x, y \in R$, $<y, x> \in \mathbb{Z}$
With $<y, x> = 2\frac{(y, x)}{(x, x)}$ and $\sigma_x(y) = y - <y, x>x$.
The first two conditions are easy to show, however I have no idea how to find the inner product $(y, x)$ given the information we have in order to show conditions three and four.
Any help appreciated, I feel like I have missed something quite obvious however I've read through my notes a few times and I can't see anything that would help.
As in the comments - but to take this question off the unanswered queue:
Use $$ (\alpha,\beta) = |\alpha|\, |\beta|\, \cos \theta.$$
As promised, though not strictly related to the question, see approx. 13 mins into
https://www.youtube.com/watch?v=SxtYFtAA3OY&list=PL5E0D6DC4BCD8309D&index=14
for a description of $G2$. If I recall correctly, the preceding lecture has a nice quick sketch of roots and co-roots, in the context of reductive groups.