  The Dirac equation in the external Coulomb field is proved to possess the
symmetry determined by the 31 operators, which form the 31-dimensional algebra.
Two different fermionic realizations of the SO(1,3) algebra of the Lorentz
group are found. Two different bosonic realizations of this algebra are found
as well. All generators of the above mentioned algebras commute with the
operator of the Dirac equation in the external Coulomb field, and, therefore,
determine the algebras of invariance of such Dirac equation. Hence, the spin
s=(1,0) Bose symmetry of the Dirac equation for the free spinor field, proved
recently in our papers, is extended here for the Dirac equation interacting
with external Coulomb field. Relativistic hydrogen atom is modeling here by
such Dirac equation. We are able to prove for the relativistic hydrogen atom
both the fermionic and bosonic symmetries known from our papers about the case
of non-interacting spinor field. New symmetry operators were found on the basis
of new gamma matrix representations of the Clifford and SO(8) algebras, which
were found recently in our papers. Hidden symmetries were found both in the
canonical Foldy--Wouthuysen and in the covariant Dirac representations. The
symmetry operators, which are simple and graceful in the Foldy--Wouthuysen
representation, become non-local in the Dirac model.
