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These notes have the pedagogical purpose of exposing mathematical accounts about of the Weyl tensor. The development of calculations is presented to obtain the curvature tensor from the conformal transformation, consequently resulting in the Weyl conformal tensor and the Riemann tensor reveals itself decomposed into its parts. The Ruse-Lanczos identity is a useful mathematical tool to build duality identities in the two pairs of antisymmetric indices of the Riemann tensor components.