  We parametrise the space of all possible flavour non-universal
$\mathfrak{u}(1)_X$ extensions of the Standard Model that embed inside
anomaly-free semi-simple gauge theories, including up to three right-handed
neutrinos. More generally, we parametrise all abelian extensions (i.e.) by any
number of $\mathfrak{u}(1)$'s) of the SM with such semi-simple completions. The
resulting 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 $\pm
10$ can be embedded in such semi-simple gauge theories. Any vector-like
anomaly-free abelian extension embeds (at least) inside $\mathfrak{g} =
\mathfrak{su}(12)\oplus \mathfrak{su}(2)_L\oplus \mathfrak{su}(2)_R$. We also
provide a simple computer program that tests whether a given
$\mathfrak{u}(1)_{X^1}\oplus \mathfrak{u}(1)_{X^2}\oplus \dots$ charge
assignment has a semi-simple completion and, if it does, outputs a set of
maximal 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 way
from $\mathfrak{sm} \oplus\mathfrak{u}(1)_{X^1}\oplus
\mathfrak{u}(1)_{X^2}\oplus \dots$ models, which have many phenomenological
uses, to their unified gauge completions in the ultraviolet.
