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I'd like to see directly from the Cartan Matrix of E8 $$\left(\begin{array}{cccccccc} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 \end{array}\right)$$ That is of dimension 248. Clearly I have that the dimension of the Cartan subalgebra is 8 so I cleary have that it is at least of dimension 24 since contains 8 copies of ${e_i,h_i,f_i}$. Is there a way to easily obtain the full dimension of the algebra just from the Cartan Matrix?

Dac0
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    The Cartan decomposition gives $\dim (E_8)=8+|\Phi|$, where $\Phi$ is the root system of $E_8$. This can be computed from the Cartan matrix, see for example here, and we obtain $|\Phi|=240$. See also this post, where the roots are constructed from the Cartan matrix in another (easier) example. – Dietrich Burde Sep 06 '22 at 11:16

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