Orthogonal matrix invariants

The orthogonal group acts on the space of several $n\times n$ matrices by simultaneous conjugation. For an infinite field of characteristic different from two, relations between generators for the algebra of invariants are described. As an application, the maximal degree of elements of a minimal system of generators is described with deviation $3$. This note contains concise but precise description of the results. All proofs can be found in arXiv: 0902.4266 and arXiv: 1011.5201.