This is in differential form notation:
d ^ R sup a sub b + omega sup a sub c ^ R sup c sub b – R sup a sub c ^ omega sup c sub b = 0
and is eq. (3.141) of Carroll’s online notes. Note carefully that it neglects torsion. UFT88 restores the torsion. In vector notation it is for spacelike indices:
del dot R sup a sub b + omega sup a sub c dot R sup c sub b – R sup a sub c dot omega sup c sub b = 0
Einstein used it as the basis for his field equation of 1915 but it is completely wrong because it neglects torsion. Obviously torsion was not known to Bianchi, and Einstein used Bianchi uncritically. In the next note I will attempt to correct it in a simple way with vector notation rather than tensor notation as in UFT88, where the correct Cartan identity was used as a starting point. The second Bianchi identity is obsolete and is no longer needed in gravitational theory because it has been replaced by the Cartan and Evans identities. Nevertheless it is very interesting to correct it as in UFT88. The latter give a vast amount of information because they have the same structure as the electromagnetic field equations. Obviously there are thousands of textbooks on electromagnetic theory, electricity and magnetism. Being just one individual, working with a small group able colleagues, I have only had time to produce a microscopic fraction of what is possible with the ECE theory of gravitation, but what has been produced is very interesting. So I must be careful and selective in what work I am going to do.