Bianchi Identities (Contracted)

Contracting $\lambda$ with $\nu$ in the Bianchi Identities

$$R_{\lambda\mu\nu\kappa;\eta}+R_{\lambda\mu\eta\nu;\kappa}+R_{\lambda\mu\kappa\eta;\nu}=0$$ (1)

gives

$$R_{\mu\kappa;\eta}-R_{\mu\eta;\kappa}+R^\nu{}_{\mu\kappa\eta;\nu}=0.$$ (2)

Contracting again,

$$R_{;\eta}-R^\mu{}_{\eta;\mu}-R^\nu{}_{\eta;\nu}=0,$$ (3)

or

$$(R^\mu{}_\eta-{\textstyle{1\over 2}}\delta^\mu{}_\eta R)_{;\mu} = 0,$$ (4)

or

$$(R^{\mu\nu}-{\textstyle{1\over 2}}g^{\mu\nu}R)_{;\mu}=0.$$ (5)