  The concept of algebroid is convenient for constructions of geometrical
frameworks. For example, metric-affine and generalized geometries can be
written on Lie and Courant algebroids, respectively. Furthermore, string
theories might make use of many other algebroids such as metric algebroids,
higher-Courant algebroids, or conformal Courant algebroids. Working on the
possibly most general algebroid structure is fruitful as it creates a chance to
study all of them at once. Local pre-Leibniz algebroids are such general ones
in which metric-connection geometries are possible to construct. Yet, the
existence of the locality operator necessitates the modification of torsion and
curvature operators to achieve tensorial quantities. In this paper, this
modification is explained from the point of view that the modification is
applied to the bracket instead. This leads one to consider `anti-commutable'
pre-Leibniz algebroids whose bracket satisfies a property defined with respect
to a choice of an equivalence class of connections. These `admissible'
connections are claimed to be the necessary ones for a geometry on algebroids
because one can prove many desirable properties for them. For instance, we
prove the first and second Bianchi identities, Ricci identity, Cartan structure
equations, the construction of Levi-Civita connections, the decomposition of
connection in terms of torsion and non-metricity. These all are possible
because the modified bracket becomes anti-symmetric for an admissible
connection so that one can apply the machinery of almost- or pre-Lie
algebroids. We investigate various algebroid structures from the literature and
show that they admit admissible connections which are metric-compatible in some
generalized sense. Moreover, we prove that local pre-Leibniz algebroids that
are not anti-commutable cannot be equipped with a torsion-free, and in
particular Levi-Civita, connection.
