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 $\mathcal{M}^#$? In this article, we answer this question completely by providing a full characterization of the structure of $\mathcal{M}$-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 $\mathcal{M}^#$. Based on these conditions, we propose several explicit design methods of specifying bent functions outside $\mathcal{M}^#$ in the special case when $f=g||h||g||(h+1)$, where $g$ and $h$ are bent functions.