  We study Faddeev formulation of gravity, in which the metric is composed of
vector fields or the tetrad of the ten-dimensional fields, $f^A_\lambda$, where
$\lambda = 1, 2, 3, 4$ and $A = 1, ..., 10$ is vector index w. r. t. the
Euclidean (or Minkowsky) ten-dimensional spacetime. We propose representation
of the type of the Cartan-Weyl one. It is based on extending the set of
variables by introducing the infinitesimal SO(10) connection. Excluding this
connection via equations of motion we reproduce the original Faddeev action. A
peculiar feature of this representation is occurrence of the local SO(10)
symmetry violating condition so that SO(10) symmetry is only global one in full
correspondence with that the original Faddeev formulation just possesses SO(10)
symmetry w. r. t. the global SO(10) rotation of the Euclidean ten-dimensional
spacetime. We also consider analog of the Barbero-Immirzi parameter which can
be naturally introduced in the considered representation.
