  In this article, we develop a further adaptation of the method of
Skjelbred-Sund to construct central extensions of axial algebras. We use our
method to prove that all axial central extensions (with respect to a maximal
set of axes) of complex simple finite-dimensional Jordan algebras are split and
that all non-split axial central extensions of dimension $n\leq 4$ over an
algebraically closed field of characteristic not $2$ are Jordan. Also, we give
a classification of $2$-dimensional axial algebras and describe some important
properties of these algebras.
