  One-sided exact categories are obtained via a weakening of a Quillen exact
category. Such one-sided exact categories are homologically similar to Quillen
exact categories: a one-sided exact category $\mathcal{E}$ can be (essentially
uniquely) embedded into its exact hull ${\mathcal{E}}^{\textrm{ex}}$; this
embedding induces a derived equivalence $\textbf{D}^b(\mathcal{E}) \to
\textbf{D}^b({\mathcal{E}}^{\textrm{ex}})$.
  Whereas it is well known that Quillen's obscure axioms are redundant for
exact categories, some one-sided exact categories are known to not satisfy the
corresponding obscure axiom. In fact, we show that the failure of the obscure
axiom is controlled by the embedding of $\mathcal{E}$ into its exact hull
${\mathcal{E}}^{\textrm{ex}}.$
  In this paper, we introduce three versions of the obscure axiom (these
versions coincide when the category is weakly idempotent complete) and
establish equivalent homological properties, such as the snake lemma and the
nine lemma. We show that a one-sided exact category admits a closure under each
of these obscure axioms, each of which preserves the bounded derived category
up to triangle equivalence.
