For orders S in cubic extensions K/F, the paper classifies precisely when S fails to embed optimally into all maximal orders of a degree-3 central simple algebra B, and when it succeeds in exactly 1/3 or 2/3 of the isomorphism classes of such maximal orders.
Zur zahlentheorie der quaternionen-algebren.Journal f¨ ur die reine und angewandte Mathematik, 195:127–151
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Optimal embeddings for maximal orders of central simple algebras of degree 3 over number fields
For orders S in cubic extensions K/F, the paper classifies precisely when S fails to embed optimally into all maximal orders of a degree-3 central simple algebra B, and when it succeeds in exactly 1/3 or 2/3 of the isomorphism classes of such maximal orders.