  We provide a Lagrangian construction for the fixed-point subalgebra, together
with its idempotent form, in a quasi-split symmetric pair of type $A_{n-1}$.
This is obtained inside the limit of a projective system of Borel-Moore
homologies of the Steinberg varieties of $n$-step isotropic flag varieties.
Arising from the construction are a basis of homological origin for the
idempotent form and a geometric realization of rational modules.
