  On the set H_n(K) of symmetric n by n matrices over the field K we can define
various binary and ternary products which endow it with the structure of a
Jordan algebra or a Lie or Jordan triple system. All these non-associative
structures have the orthogonal Lie algebra so(n,K) as derivation algebra. This
gives an embedding of so(n,K) into so(N,K) for N = n(n+1)/2 - 1. We obtain a
sequence of reductive pairs (so(N,K), so(n,K)) that provides a family of
irreducible Lie-Yamaguti algebras. In this paper we explain in detail the
construction of these Lie-Yamaguti algebras. In the cases n < 5, we use
computer algebra to determine the polynomial identities of degree < 7; we also
study the identities relating the bilinear Lie-Yamaguti product with the
trilinear product obtained from the Jordan triple product.
