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.

Tetradic Palatini action

which, after multiplication by $e\; \{I\; \backslash beta\}$ just tells us that the Einstein tensor $R\; \{\backslash alpha\; \backslash beta\}\; -\; \{1\; \backslash over\; 2\}\; R\; g\; \{\backslash alpha\; \backslash beta\}$ of the metric defined by the tetrads vanishes.

see also

Contributors to the mathematical background for general relativity

Isaac Newton (Newton's identities for characteristic of Einstein tensor)