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We parametrise the space of all possible flavour non-universal$\mathfrak{u}(1)_X$ extensions of the Standard Model that embed insideanomaly-free semi-simple gauge theories, including up to three right-handedneutrinos. More generally, we parametrise all abelian extensions (i.e.) by anynumber of $\mathfrak{u}(1)$'s) of the SM with such semi-simple completions. Theresulting space of abelian extensions is a collection of planes of dimensions$\leq 6$. Numerically, we find that roughly $2.5\%$ of anomaly-free$\mathfrak{u}(1)_X$ extensions of the SM with a maximum charge ratio of $\pm10$ can be embedded in such semi-simple gauge theories. Any vector-likeanomaly-free abelian extension embeds (at least) inside $\mathfrak{g} =\mathfrak{su}(12)\oplus \mathfrak{su}(2)_L\oplus \mathfrak{su}(2)_R$. We alsoprovide a simple computer program that tests whether a given$\mathfrak{u}(1)_{X^1}\oplus \mathfrak{u}(1)_{X^2}\oplus \dots$ chargeassignment has a semi-simple completion and, if it does, outputs a set ofmaximal gauge algebras in which the$\mathfrak{sm}\oplus\mathfrak{u}(1)_{X^1}\oplus \mathfrak{u}(1)_{X^2}\oplus\dots$ model may be embedded. We hope this is a useful tool in pointing the wayfrom $\mathfrak{sm} \oplus\mathfrak{u}(1)_{X^1}\oplus\mathfrak{u}(1)_{X^2}\oplus \dots$ models, which have many phenomenologicaluses, to their unified gauge completions in the ultraviolet.

Flatland: abelian extensions of the Standard Model with semi-simple completions