  String and M theories seem to require generalizations of usual notions of
differential geometry. Such generalizations usually involve extending the
tangent bundle to larger vector bundles equipped with various algebroid
structures. The most general geometric scheme is not well understood yet, and a
unifying framework for such algebroid structures is needed. Our aim in this
paper is to propose such a general framework. Our strategy is to follow the
hierarchy of defining axioms for a Courant algebroid: almost-Courant - metric -
pre-Courant - Courant. In particular, we focus on the symmetric part of the
bracket and the metric invariance property, and try to make sense of them in a
manner as general as possible. These ideas lead us to define new algebroid
structures which we dub Bourbaki and metric-Bourbaki algebroids. For a special
case of exact pre-metric-Bourbaki algebroids, we construct maps which
generalize the Cartan calculus of exterior derivative, Lie derivative and
interior product. This is done by a reverse-mathematical analysis of Severa
classification of exact Courant algebroids. Abstracting crucial properties of
these maps, we define the notion of Bourbaki pre-calculus. Conversely, given a
Bourbaki pre-calculus, we construct a pre-metric-Bourbaki algebroid with a
standard bracket analogous to Dorfman bracket. We prove that any exact
pre-metric-Bourbaki algebroid satisfying certain conditions has to have a
bracket that is the twisted version of the standard bracket. We prove that many
algebroids from the literature are examples of these new algebroids. One
straightforward generalization of our constructions might be done by replacing
the tangent bundle with a Lie algebroid A. This step allows us to define
A-Bourbaki algebroids and Bourbaki A-pre-calculus, and extend our results,
while proving many other algebroids from the literature fit into this
framework.
