In 1968 [1], Ernst derived his famous equation for a single complex variable (which acts as a potential for the metric components) for the Einstein equations. The Lagrangian he writes down, just above equation (4) in [1], seems to be very non-trivial to derive. I had thought it was just the Einstein-Hilbert Lagrangian $\sqrt{-g} R$, but it is not since $R$ contains second order derivatives which do not appear in Ernst's Lagrangian. However, they are suspiciously similar (the terms involving $\omega$ are exactly the same).

Can someone please explain how the derivation goes? Is it the Einstein-Hilbert Lagrangian but modified by using the actual field equations themselves to remove some terms? Or do some other tricky manipulations take place?

Given Ernst's Lagrangian I can derive the rest of his equations quite nicely, following through the Euler-Lagrange approach. It is just the actual Lagrangian itself I cannot reproduce!

Thanks

[1] Ernst, 1968, "New formulation of the axially symmetric gravitational field problem"