X-Nico

Holst action

Variation of the first term of the action with respect to the tetrad $e^\{\backslash alpha\}\; \{\backslash \; I\}$ gives the (mixed index) Einstein tensor and variation of the second term with respect to the tetrad gives a quantity that vanishes by symmetries of the Riemann tensor (specifically the first Bianchi identity), together these imply Einstein's vacuum field equations hold.

see also