We express any Courant algebroid bracket by means of a metric connection, and
construct a Courant algebroid structure on any orthogonal Whitney sum E⊕C where E is a given Courant algebroid and C is a flat, pseudo- Euclidean
vector bundle. Then, we establish the general expression of the bracket of a
transitive Courant algebroid, i.e., a Courant algebroid with a surjective
anchor, and describe a class of transitive Courant algebroids which are Whitney
sums of a Courant subalgebroid with neutral metric and Courant-like bracket and
a pseudo-Euclidean vector bundle with a flat, metric connection. In particular,
this class contains all the transitive Courant algebroids of minimal rank; for
these, the flat term mentioned above is zero. The results extend to regular
Courant algebroids, i.e., Courant algebroids with a constant rank anchor. The
paper ends with miscellaneous remarks and an appendix on Dirac linear spaces.Comment: LaTex, 27 page