In recent years, the work of Druel, Guenancia, Greb, Kebekus, Höring, and Peternell has shown that an analogue of the Beauville–Bogomolov (BB) decomposition holds for K-trivial varieties with klt singularities over the complex numbers. Motivated by the MMP and log CY geometry, it is natural to ask whether the BB decomposition can be extended to the log canonical setting. In this talk, I will provide a negative answer by constructing, for each integer d>3, a K-trivial log canonical variety of dimension d that does not admit a BB-type decomposition. More precisely, we prove its Albanese morphism is not birationally isotrivial (so the abelian part cannot be split off after an étale cover). This is joint work with S. Filipazzi, Zs. Patakfalvi, N. Tsakanikas, and N. Müller.