  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}.
