  A finite-dimensional Lie algebra $L$ over a field $F$ is called an
$A$-algebra if all of its nilpotent subalgebras are abelian. This is analogous
to the concept of an $A$-group: a finite group with the property that all of
its Sylow subgroups are abelian. These groups were first studied in the 1940s
by Philip Hall, and are still studied today. Rather less is known about
$A$-algebras, though they have been studied and used by a number of authors.
The purpose of this paper is to obtain more detailed results on the structure
of solvable Lie $A$-algebras. It is shown that they split over each term in
their derived series. This leads to a decomposition of $L$ as $L = A_{n}
\dot{+} A_{n-1} \dot{+} ... \dot{+} A_0$ where $A_i$ is an abelian subalgebra
of $L$ and $L^{(i)} = A_{n} \dot{+} A_{n-1} \dot{+} ... \dot{+} A_{i}$ for each
$0 \leq i \leq n$. It is shown that the ideals of $L$ relate nicely to this
decomposition: if $K$ is an ideal of $L$ then $K = (K \cap A_n) \dot{+} (K \cap
A_{n-1}) \dot{+} ... \dot{+} (K \cap A_0)$. When $L^2$ is nilpotent we can
locate the position of the maximal nilpotent subalgebras: if $U$ is a maximal
nilpotent subalgebra of $L$ then $U = (U \cap L^2) \oplus (U \cap C)$ where $C$
is a Cartan subalgebra of $L$. If $L$ has a unique minimal ideal $W$ then $N =
Z_L(W)$. If, in addition, $L$ is strongly solvable the maximal nilpotent
subalgebras of $L$ are $L^2$ and the Cartan subalgebras of $L$ (that is, the
subalgebras that are complementary to $L^2$.) Necessary and sufficient
conditions are given for such an algebra to be an $A$-algebra. Finally, more
detailed structure results are given when the underlying field is algebraically
closed.
