How to represent a conformal transformation using spinors?

In my mathematical meanderings, I’ve come across a few references to conformal transformations being representable with spinors. Now generally I think of a conformal transformation in a Riemanninan space with metric as being of the from:

$ $ g_{\mu\nu}\longrightarrow\lambda^{2}g_{\mu\nu}$ $

Where $ \lambda$ may be a function of the coordinates. In general I understand you can represent a conformal transformation as a composition of this dilation (our $ \lambda)$ with a rotation (lets call that $ R$ ) which in component form looks like:

$ $ \tilde{g}_{\alpha\beta}=\lambda R_{\alpha}^{\mu}g_{\mu\nu}R_{\beta}^{\nu}\lambda=\lambda R^{T}gR\lambda$ $

Where the latter term is written in matrix form. So how do we get from here to a spinor representation of our transformation??? Note I’m just looking at conformal transformations of $ R^{3,1}$ (with a point at infinity to compactify it).