  $SL^\infty$ denotes the space of functions whose square function is in
$L^\infty$, and the subspaces $SL^\infty_n$, $n\in\mathbb{N}$, are the finite
dimensional building blocks of $SL^\infty$.
  We show that the identity operator $I_{SL^\infty_n}$ on $SL^\infty_n$ well
factors through operators $T : SL^\infty_N\to SL^\infty_N$ having large
diagonal with respect to the standard Haar system. Moreover, we prove that
$I_{SL^\infty_n}$ well factors either through any given operator $T :
SL^\infty_N\to SL^\infty_N$, or through $I_{SL^\infty_N}-T$. Let $X^{(r)}$
denote the direct sum $\bigl(\sum_{n\in\mathbb{N}_0} SL^\infty_n\bigr)_r$,
where $1\leq r \leq \infty$. Using Bourgain's localization method, we obtain
from the finite dimensional factorization result that for each $1\leq r\leq
\infty$, the identity operator $I_{X^{(r)}}$ on $X^{(r)}$ factors either
through any given operator $T : X^{(r)}\to X^{(r)}$, or through $I_{X^{(r)}} -
T$. Consequently, the spaces $\bigl(\sum_{n\in\mathbb{N}_0}
SL^\infty_n\bigr)_r$, $1\leq r\leq \infty$, are all primary.
