  Electroencephalographic signals are represented as multidimensional datasets.
We introduce an enhancement to the augmented covariance method (ACM),
exploiting more thoroughly its mathematical properties, in order to improve
motor imagery classification.Standard ACM emerges as a combination of phase
space reconstruction of dynamical systems and of Riemannian geometry. Indeed,
it is based on the construction of a Symmetric Positive Definite matrix to
improve classification. But this matrix also has a Block-Toeplitz structure
that was previously ignored. This work treats such matrices in the real
manifold to which they belong: the set of Block-Toeplitz SPD matrices. After
some manipulation, this set is can be seen as the product of an SPD manifold
and a Siegel Disk Space.The proposed methodology was tested using the MOABB
framework with a within-session evaluation procedure. It achieves a similar
classification performance to ACM, which is typically better than -- or at
worse comparable to -- state-of-the-art methods. But, it also improves
consequently the computational efficiency over ACM, making it even more
suitable for real time experiments.
