Аннотация:
Tits construction produces a Lie algebra out of a composition algebra and an exceptional Jordan algebra. The type of the result is $F_4$, ${}^2E_6$, $E_7$ or $E_8$ when the composition algebra has dimension 1,2,4 or 8 respectively. Garibaldi and Petersson noted that the Tits index ${}^2E_6^{35}$ cannot occur as a result of Tits construction. Recently Alex Henke proved that the Tits index $E_7^{66}$ is also not possible. We push the analogy further and show that Lie algebras of Tits index $E_8^{133}$ don't lie in the image of the Tits construction. The proof relies on basic facts about symmetric spaces and our joint result with Garibaldi and Semenov about isotropy of groups of type $E_7$ in terms of the Rost invariant. This is a part of a work joint with Simon Rigby.