# How do I find all the roots of a lie algebra and hence its root system diagram given the cartan matrix for that algebra?

If I am given the cartan matrix, I can find the $2\langle αi , αj\rangle/\langle αj , αj\rangle$ of the simple roots where $α$ are the simple roots. But, from this how do I find the $\langle αi , αj\rangle$ themselves. After finding this, how do I find the other roots and their scalar products, and hence construct the root diagram?

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The lengths of the roots can be determined from the Cartan matrix $A_{ij}=2\frac{(\alpha_i, \alpha_j)}{(\alpha_j, \alpha_j)}$ up to an overall normalization.
$\frac{A_{ij}}{A_{ji}} = \frac{(\alpha_i, \alpha_i)}{(\alpha_j, \alpha_j)}$