  We provide a characterization of homogeneous spaces under a reductive group
scheme such that the geometric stabilizers are maximal tori. The quasi-split
case over a semilocal base is of special interest and permits to answer a
question raised by Marc Levine on homogeneous SL$_n$-spaces. At the end, we
provide an application to the local-global principles for embeddings of
{\'e}tale algebras with involution into central simple algebras with
involution.
