We will show that any self-adjoint Latin subsquare with constant diagonal can be extended to a Latin square with the same property. As a consequence, every loop with inverses satisfying the identity (xy)−1 = y −1 x−1 (an IAA loop for short) is locally embed- dable into nite IAA loops, and its loop algebra is locally embeddable into loop algebras of nite IAA loops. The IAA property enables to extend this result to loop algebras with the natural involution arising from the inverse map on the loop. In particular, this is true for groups and their group algebras.