wikipedia doesn't say, nor my Berger Panorama book (but I might google Levi-Civita to get rid of one level of brackets) and the library is far (actually not, but it has German Schließungszeiten and I got no key). (I guess the differential Bianchi identity is not due to Bianchi? So who did which?)
To clarify on the "vice versa": according to my Swiss cheese memory, there might be something completely different actually due to Bianchi. Some book was telling this quite nice…
Said MO answer is here
Best Answer
L.P. Eisenhart claims in his Riemannian Geometry (Princeton University Press, 1926, p. 82) that Bianchi was the first who discovered the algebraic identity in 1902.
It is also indicated in Italian Mathematics Between the Two World Wars by Guerraggio and Nastasi (Birkhäuser Verlag, 2006, p. 15) that Bianchi obtained both of the identities which were published in his paper "Sui simboli di Riemann a quattro indici e sulla curvatura di Riemann", Rend. Acc. Lincei, 11 (1902), pp. 3–7.