**Abstract: ** We shall review our research program the aim of which is to find
all indecomposable solvable extensions of a given class of nilpotent Lie
algebras. Specifically in this talk we consider a nilpotent Lie algebra
$\mathfrak{n}$ that is isomorphic to the nilradical of the Borel subalgebra
of a complex simple Lie algebra, or of its split real form. We treat all
classical and exceptional simple Lie algebras in a uniform manner. We
identify the nilpotent Lie algebra $\mathfrak{n}$ as the one consisting of
all positive root spaces. We present general structural properties of all
solvable extensions of $\mathfrak{n}$. In particular, we study the extension
by one nonnilpotent element and by the maximal number of such elements. We
show that the extension of maximal dimension is always unique and isomorphic
to the Borel subalgebra of the corresponding simple Lie algebra.
(In collaboration with Pavel Winternitz, J. Phys. A: Math. Theor. 45 (2012)
095202.)