  We obtain the Arnowitt-Deser-Misner formulation of general relativity in $n$
dimensions ($n \geq 3$) from its either $SO(n-1,1)$ [$SO(n)$] or $SO(n-1)$
Palatini Hamiltonian formulations and vice versa [we recall that $SO(n-1,1)$
[$SO(n)$] requires no gauge fixing whereas $SO(n-1)$ involves the time gauge].
Similarly, the Hamiltonian formulation of general relativity in terms of
Ashtekar-Barbero variables can also be directly obtained from the
Arnowitt-Deser-Misner Hamiltonian formulation and vice versa, which is an
alternative approach to the way followed by Barbero. We give the relevant maps
among the phase-space variables and relate the corresponding symplectic
structures and the first-class constraints.
