  Gradients are natural first order differential operators depending on
Riemannian metrics. The principal symbols of them are related to the enveloping
algebra and higher Casimir elements. We give certain relations in the
enveloping algebra, which induce not only identities for higher Casimir
elements but also all Bochner-Weitzenb\"ock formulas for gradients. As
applications, we give some vanishing theorems.
