finite-dimensional vector space endowed with a non-degenerate quadratic form (in case of an euclidean space, a positive definite one).

The ${\EuScript{J}}$-algebra here means a pseudo-euclidean structure given by a $C^1$ non-degenerate quadratic form ${\EuScript{J}}$, defined on $\Lambda$, which generates positive and negative cones with maximal dimension $p$ and $q$, respectively, with $p + q = \dim (M)$.

The maximal dimension of a cone in $T_xM$ is the maximal dimension of the subspaces contained in there.

This algebraic/geometric approach has been very useful in the study of weak and uniform hyperbolicity, see [@BurnKatok94], [@lewow80], [@Wojtk01]. In [@ArSal2012], this author jointly with V. Araújo, obtained characterizations of partial and singular/sectional hyperbolicity based on ${\EuScript{J}}$-algebra. In [@ArSal2015], the same authors proved an equivalence between dominated splittings for the flow and dominated splittings for the $k$-th exterior powers of the tangent cocycle.

More results relating geometric and algebraic features of singular hyperbolicity can be viewed in [@ArSal2012], [@ArSal2015], [@SalgCoelho2017], for the classical sectional and singular hyperbolicity definitions. Also see [@Salg2016] for singular hyperbolicity in a broad sense involving sectional expansion of intermediate dimensions between two and the full dimension of the central subbundle.

Our main result (Theorem A) rounds about of the so called 