The spinor representation formula in 3 and 4 dimensions

In the literature, two approaches to the Weierstrass representation formula using spinors are known, one explicit, going back to Kusner & Schmitt, and generalized by Konopelchenko and Taimanov, and one abstract due to Friedrich, Bayard, Lawn and Roth. In this article, we show that both points of view are indeed equivalent, for surfaces in R^3, Nil_3 and R^4. The correspondence between the equations of both approaches is explicitly given and as a consequence we derive alternate (and simpler) proofs of these previous theorems.