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A real polynomial P of degree n in one real variable is hyperbolic if its roots are  all real. A real-valued function P is called a hyperbolic polynomial-like function  (HPLF) of degree n if it has n real zeros and P (n) vanishes nowhere. Denote  (i)  by xk the roots of P (i) , k = 1, . . . , n - i, i = 0, . . . , n - 1. Then in the absence  of any equality of the form  (j ) (l)  xi = xk (1)  one has  (i)  (j )  (i)  (2)  xk < xk < xk  i < j +j -i   (the Rolle theorem). For n 4 (resp. for n 5) not all arrangements without  (i)  equalities (1) of n(n + 1)/2 real numbers xk and compatible with (2) are real izable by the roots of hyperbolic polynomials (resp. of HPLFs) of degree n and  ( 1) ( 1)  ( 3) ( 3)  ( 1) ( 1)  of their derivatives. For n = 5 and when x1 < x2 < x1 < x2 < x3 < x4  we show that from the 40 arrangements without equalities (1) and compatible  with (2) only 16 are realizable by HPLFs (from which 6 by perturbations of  hyperbolic polynomials and none by hyperbolic polynomials).In this work the properties of Cartan subalgebras and weight spaces of finite dimensional Lie algebras are extended to the case of Leibniz algebras. Namely, the relation between Cartan subalgebras and regular elements are described, also an analogue of Cartan’s criterion of solvability is proved

Root Arrangements of Hyperbolic Criterion of Solvability of Finite Dimensional Leibniz Algebras