Symmetric spaces, Lie triple systems and a construction of Rosenfeld planes

It is well known that symmetric spaces are in one-to-one correspondence with Lie triple systems. On the other hand, works of Boris Rosenfeld and a recent result of Huang and Leung show that symmetric spaces may be viewed as “Rosenfeld planes” (a generalization of the Cayley plane over the tensor product of two alternative algebras) or with certain Grassmanninans related to them. Also it is known that the sum of two copies of the tensor product of two alternative algebras carries a structure of a Lie triple system. This hints that there should be a direct construction of the Rosenfeld plane starting from a Lie triple system. In the talk such a construction is proposed.