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I've heard that one can construct the exceptional Lie algebra $E_8$ as the Lie algebra of the group of isometries of projective plane over octonions, or something of this form. Unfortunately, I do not remember the exact statement.

Can you, please, explain how to construct $E_8$ using octonions? Or maybe you know a good reference with this model of $E_8$?

Are there any other nice models for $E_8$?

I know that one can use cyclic $\mathbb{Z}/3\mathbb{Z}$-grading on $E_8$ to get decomposition $E_8=\wedge^3(V^*)\oplus sl(V)\oplus \wedge^3(V)$, where $\dim V=9$. But the problem is that the bracket on such decomposition is defined in a very tricky way, so I don't think this can be regarded as an honest model for $E_8$.

Thank you very much!

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up vote 1 down vote accepted

There are so called Cayley integers that form an $E_8$ lattice in the space of octonions. There was a long disscusion of integral octonions in n-category cafe see here: http://golem.ph.utexas.edu/category/2013/07/integral_octonions_part_1.html

There is a disscusion of the associated lie algebras constructions, I believe $E_8$ is also constructed there. Note, that this link is to the first out of six parts.

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