Tits construction and the Rost invariant

Simple Lie algebras over an algebraically closed field of characteristic 0 are described by Dynkin diagrams. Over a non-closed field, the same Dynkin diagram can correspond to many simple algebras, so it is interesting to study constructions of simple Lie algebras and invariants that make it possible to recognize their isomorphism or reflect some of their properties. One such construction of exceptional (i.e., types $E_6$, $E_7$, $E_8$, $F_4$ or $G_2$) Lie algebras was proposed by Jacques Tits; the Jordan algebra and an alternative algebra are given as input, and the output is a Lie algebra, and all real forms of Lie algebras can be constructed in this way. One of the most useful invariants (with meaning in the third Galois cohomology group) was constructed by Markus Rost. We show that a Lie algebra of (outer) type $E_6$ is obtained by the Tits construction if and only if the Rost invariant is a pure symbol. As an application of this result we prove a Springer-type theorem for an $E_6$-homogeneous manifold.