Let $\mathrm{ Lie} \left(n, k\right)$ denote the class of all $n$-dimensional real solvable Lie algebras having $k$-dimensional derived ideal ($1 \leqslant k \leqslant n-1$). In 1993, the class $\mathrm{ Lie} \left(n, 1\right)$ was completely classified by Sch\"obel \cite{ Sch93}. In 2016, Vu A. Le et al. \cite{ VHTHT16} considered the class $\mathrm{ Lie} \left(n, n-1\right)$ and classified its subclass containing all the algebras having 1-codimensional commutative derived ideal. One subclass in { \Li} was firstly considered and incompletely classified by Sch\"obel \cite{ Sch93} in 1993. Later, Janisse also gave an incomplete classification of { \Li} and published as a scientific report \cite{ Jan10} in 2010. In this paper, we set up a new approach to study the classifying problem of classes { \Li} as well as { \li} and present the new complete classification of { \Li} in the combination with the well-known Eberlein's result of 2-step nilpotent Lie algebras from \cite[p.\,37--72]{ Ebe03}. The paper will also classify a subclass of { \li} and will point out missings in Sch\"obel \cite{ Sch93}, Janisse \cite{ Jan10}, Mubarakzyanov \cite{ Mub63a} as well as revise an error of Morozov \cite{ Mor58}.
Categories: math.RA