I have a set of vectors in the Euclidean space of dimension 4 (12 vectors),and I would like to know how can prove whether this set is a root system of Lie algebra or not?
$ +/-{(e_i-e_j) } \space for \space {i\neq j}$
Where $\space{e_i\cdot e_j =\delta_{ij}} \space for \space i,j=1,2,3,4$
Note: $(e_i-e_j)\cdot \sum_{k=1}^4 e_k = 0 $
Properties:
Let $\Phi$ be a set of vectors
1 - $\Phi$ does not contain zero, occupies a Euclidean space of the same dimension that classifies Lie G algebra and the number of elements of $\Phi$ is equal to dim , G - rank , G
2 - If $\alpha$ ; $\epsilon$ ; $\Phi$, the only multiples of $\alpha$ in $\Phi$ are +/- $\alpha$
3 - If $\alpha$, $\beta$ , $\epsilon$ ; $\Phi$, so $\frac{2(\alpha \cdot\beta)}{\alpha^{2}}$ is an integer.
4 - If $\alpha$ , $\beta$ $\epsilon$ $\Phi$, then $\sigma_{\alpha}(\beta)$ exist in $\Phi$, that is, Weyl's group leaves $\Phi$ invariable