Stephan Maier, in 1998, studied the conformal flatness of 3 and 4-dimensional Thurston geometries. The second author with R. Deszcz and L. Verstraelen proved, in 2006, that every 3-dimensional Thurston geometry is pseudo-symmetric (in the sense of Deszcz). We prove   that except the symmetric 4-dimensional Thurston geometries, any 4-dimensional Thurston geometry is neither Ricci pseudo-symmetric nor Weyl pseudo-symmetric. We also show that  (the only non- symmetric 4-dimensional Thurston geometry which has a Kählerian structure) is holomorphically pseudo-symmetric of constant type.

pseudo-symmetry, Ricci pseudo-symmetry, Weyl pseudo-symmetry, Thurston geometry.