A Pseudoconfiguration of Points without Adjoint

We give an example of a simple oriented matroid D that admits an oriented adjoint. Furthermore, any adjoint of the underlying matroidD, does not itself admit an adjoint.Darises from the well-known non-Desargues matroid by a coextension by a coparallel element and, hence, has rank 4. The orientability ofDand some of its adjoints follows from an apparently new oriented matroid construction given in the paper that is a very special case of an amalgam of two copies of one oriented matroid.