On the set Hn(K) of symmetric n × 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 nonassociative structures have the orthogonal Lie algebra so(n, K) as derivation algebra. This gives an embedding so(n, K) ⊂ so(N, K) for 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 ≤ 4, we use computer algebra to determine the polynomial identities of degree ≤ 6; we also study the identities relating the bilinear Lie-Yamaguti product with the trilinear product obtained from the Jordan triple product.