  We construct relativistic-invariant spinning-particle Lagrangian without
auxiliary variables. Spin is considered as a composed quantity constructed on
the base of non-Grassmann vector-like variable. The variational problem
guarantees both fixed value of spin and Frenkel condition on spin-tensor.
Taking into account the Frenkel condition, we obtain, inevitably, relativistic
corrections to the algebra of position variables: their classical brackets
became noncommutative, with the "parameter of non-commutativity" proportional
to the spin-tensor. This leads to a number of interesting consequences in
quantum theory. We construct the relativistic quantum mechanics in canonical
formalism (in physical-time parametrization) and in covariant formalism (in
arbitrary parametrization). We show how state-vectors and operators of
covariant formulation can be used to compute mean values of physical operators
of position and spin. This proves relativistic covariance of canonical
formalism. Various candidates for position and spin operators of an electron
acquire clear meaning and interpretation in the Lagrangian model of Frenkel
electron. We also establish the relation between Frenkel electron and
positive-energy sector of Dirac equation, this allowed us to turn to the
long-standing problem on spin and position operators of Dirac theory. Contrary
to widely assumed opinion, our results argue in favor of Pryce's (d)-type
operators. This implies that effects of non-commutativity could be presented at
the Compton wave length, in contrast to conventional expectations at the Planck
length. At last, we present the manifestly covariant form of spin and position
operators of Dirac equation.
