When does a bent concatenation not belong to the completed Maiorana-McFarland class?

Every Boolean bent function f can be written either as a concatenation f=f₁||f₂ of two complementary semi-bent functions f₁,f₂; or as a concatenation f=f₁||f₂||f₃||f₄ of four Boolean functions f₁,f₂,f₃,f₄, all of which are simultaneously bent, semi-bent, or 5-valued spectra-functions. In this context, it is essential to ask: When does a bent concatenation f (not) belong to the completed Maiorana-McFarland class M^#? In this article, we answer this question completely by providing a full characterization of the structure of M-subspaces for the concatenation of the form f=f₁||f₂ and f=f₁||f₂||f₃||f₄, which allows us to specify the necessary and sufficient conditions so that f is outside M^#. Based on these conditions, we propose several explicit design methods of specifying bent functions outside M^# in the special case when $f=g||h||g||(h+1)$, where g and h are bent functions.