  We review Holst formalism and we discuss dynamical equivalence with standard
GR (in dimension 4). Holst formalism is written for a spin coframe field
$e^I_\mu$ and a $Spin(3,1)$-connection $\omega^{IJ}_\mu$ on spacetime $M$ and
it depends on the Holst parameter $\gamma\in \mathbb{R}-\{0\}$.
  We show the model is dynamically equivalent to standard GR, in the sense that
up to a pointwise $Spin(3,1)$-gauge transformation acting on frame indices,
solutions of the two models are in one-to-one correspondence. Hence the two
models are classically equivalent.
  One can also introduce new variables by splitting the spin connection into a
pair of a $Spin(3)$-connection $A^i_\mu$ and a $Spin(3)$-valued 1-form
$k^i_\mu$. The construction of these new variables relies on a particular
algebraic structure, called a reductive splitting. A reductive splitting is a
weaker structure than requiring that the gauge group splits as the products of
two sub-groups, as it happens in Euclidean signature in the selfdual
formulation originally introduced in this context by Ashtekar, and it still
allows to deal with the Lorentzian signature without resorting to
complexifications.
  The reductive splitting of $SL(2, \mathbb{C})$ is not unique and it is
parameterized by a real parameter $\beta$, called the Immirzi parameter. The
splitting is here done on spacetime, not on space, to obtain a
$Spin(3)$-connection $A^i_\mu$, which is called the Barbero-Immirzi connection
on spacetime. One obtains a covariant model depending on the fields $(e^I_\mu,
A^i_\mu, k^i_\mu)$ which is again dynamically equivalent to standard GR (as
well as the Holst action).
  Usually, in the literature one sets $\beta=\gamma$ for the sake of
simplicity. Here we keep the Holst and Immirzi parameters distinct to show that
eventually, only $\beta$ will survive in boundary field equations.
