Algebraic simplifications in supergravity theories

Supercovariant derivatives and supercovariant field strengths are used to put the fermionic equations of motion derived from supergravity theories in an elegant form. These results apply also to the SO(4) extended supergravity theory which is presented here. We discuss how to obtain more compact proofs of the invariance of the supergravity actions already known. In particular for the SO(2) theory a set of new auxilary fields is introduced which linearizes the equation of motion of the $\text{spin}\text{-}\text{3}\text{2}$ field.