  There is renewed interest in the coideal subalgebras used to form quantum
symmetric pairs because of recent discoveries showing that they play a
fundamental role in the representation theory of quantized enveloping algebras.
However, there is still no general theory of finite-dimensional modules for
these coideals. In this paper, we establish an important step in this
direction: we show that every quantum symmetric pair coideal subalgebra admits
a quantum Cartan subalgebra which is a polynomial ring that specializes to its
classical counterpart. The construction builds on Kostant and Sugiura's
classification of Cartan subalgebras for real semisimple Lie algebras via
strongly orthogonal systems of positive roots. We show that these quantum
Cartan subalgebras act semisimply on finite-dimensional unitary modules and
identify particularly nice generators of the quantum Cartan subalgebra for a
family of examples.
