  A \textbf{strong arc decomposition} of a (multi-)digraph $D(V, A)$ is a
partition of its arc set $A$ into two disjoint arc sets $A_1$ and $A_2$ such
that both of the spanning subdigraphs $D(V, A_1)$ and $D(V, A_2)$ are strong.
In this paper, we fully characterize all split digraphs that do not have a
strong decomposition. This resolves two problems proposed by Bang-Jensen and
Wang and contributes to a series of efforts aimed at addressing this problem
for specific graph classes. This work continues the research on semicomplete
composition [Bang-Jensen, Gutin and Yeo, J. Graph Theory, 2020]; on locally
semicomplete digraphs [Bang-Jensen and Huang, J. Combin. Theory Ser. B, 2010];
on a type of tournaments [Bang-Jensen and Yeo, Combinatorica, 2004].
