  Unextendible sets of Mutually Unbiased Bases (MUBs) are examined from the
point of view of complementary subalgebras. We show, that the linear span of
less than $d+1$ factors of $M_d \otimes M_d$ does not contain pure states, and
therefore some complementary decompositions give rise to undextendable sets of
MUBs. We provide some new complementary decompositions, and thus prove strong
unextendibility of some set of MUBs.
