Planes of symmetry of a molecule

Are we limited to use only xy, yz and zx planes to check for planes of symmetry of a molecule? Because I see a plane of symmetry in the following molecule through the two shown H atoms or OH groups. I am aware of the 3D structure shown by this projection. And since this molecule is Chiral, and has enantiomeric mirror image, this existence of a plane of symmetry appears contradictory.

• This question would be more suited to Chemistry? – joseph h Feb 16 at 7:10
• Mother Nature uses whatever axes she wants, and is not limited to our frame of reference, so, yes. – Jon Custer Feb 16 at 14:31
• Isn't this the meso isomer of this chemical? I'll admit that I'm rusty on what 3D structure is understood by this diagram, but if the main carbon chain is "zig-zagging" towards & away from the viewer, then the molecule is superposable on itself via a point reflection about the center of mass (and therefore via a plane reflection as well.) – Michael Seifert Feb 16 at 18:36
• @MichaelSeifert No, this is not a meso, and no, it has neither point nor plane symmetry. – Ivan Neretin Feb 16 at 23:23
• I think both answers so far fail to address properly the utter otherworldliness of the premise in question. You have a plane of symmetry (approximate, of course); are you only allowed to walk along the xy, yz and zx planes of the coordinate system which is official in your jurisdiction? I guess no. Well, neither are molecules. – Ivan Neretin Feb 16 at 23:43

To answer the general part of the question: no, we are not limited to the Cartesian planes and axes. The obvious examples are the ammonia and the methane milecules: the former has several reflection planes, positioned at $$120^\circ$$, whereas the latter has multiple reflection planes and also the rotation axes (one can easily google both).
@gryphus's answer is correct for the statement that there is no mirror plane in the structure of the question. Independent that this is a late comment to the answer, the reader of such a Fischer projection should not hesitate to construct a model of the underlying structure (here obviously with quite some non favourable steric interactions) to get the molecule a 3D object: