  We introduce 3+1 timelike foliation of the four dimensional Lorentz manifold
to derive the 3+1 Sen-Ashtekar-Barbero-Immirzi formalism in case of $SO(2,1)$
rotation gauge group, which is possible due to the existence of the $so(2,1)$
algebra isomorphism to $R^3_{2,1}$ algebra with respect to the vector product.
We prove that the newly introduced flux and extrinsic curvature variables
preserve the symplectic structure of the original variables. We then introduce
the modified rotational constraint and succeed to write it as a Gauss
constraint of a newly obtained connection. The newly obtained connection is
slightly different from the classical 3+1 spacelike
Sen-Ashtekar-Barbero-Immirzi connection as it contains in addition the
Minkowski metric $\eta_{ij}$ as a coefficient. Our result has a very simple
form and clearly shows how $so(2,1)$ connection is different from $so(3)$ one.
Also it is the first time that the key-stone fact that makes the whole
formalism work in timelike 3+1 case, i.e. $so(2,1) \simeq R^3_{2,1}$
isomorphism and its relation to the $so(2,1)$ connection has been researched.
