Direct sums of finite dimensional $SL^infty_n$ spaces

$SL^infty$ denotes the space of functions whose square function is in $L^infty$, and the subspaces $SL^infty_n$, $ninmathbb{N}$, 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 $T : SL^infty_Nto SL^infty_N$ 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 $T : SL^infty_Nto SL^infty_N$, or through $I_{SL^infty_N}-T$. Let $X^{(r)}$ denote the direct sum $bigl(sum_{ninmathbb{N}_0} SL^infty_nbigr)_r$, where $1leq r leq infty$. Using Bourgains localization method, we obtain from the finite dimensional factorization result that for each $1leq rleq infty$, 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 $bigl(sum_{ninmathbb{N}_0} SL^infty_nbigr)_r$, $1leq rleq infty$, are all primary.