  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.
