  In recent work where Matsusaka generalizes the relationship between
Habiro-type series and false theta functions after Hikami, five families of
Hecke-type double-sums of the form \begin{equation*} \left( \sum_{r,s\ge 0
}-\sum_{r,s<0}\right)(-1)^{r+s}x^ry^sq^{a\binom{r}{2}+brs+c\binom{s}{2}},
\end{equation*} where $b^2-ac<0$, are decomposed into sums of products of theta
functions and false theta functions. Here we obtain a general formula for such
double-sums in terms of theta functions and false theta functions, which
subsumes the decompositions of Matsusaka. Our general formula is similar in
structure to the case $b^2-ac>0$, where Mortenson and Zwegers obtain a
decomposition in terms of Appell functions and theta functions.
