I'm trying to decompose the Kazhdan-Lusztig C' basis element associated to the longest word in $S_n$, $C'_{w_0}$ into products and sums of elements $C'_w$ where $w < w_0$ in the Bruhat order. For example I know that $C'_{s_1s_2s_1}=C'_{s_1}C'_{s_2}C'_{s_1}-C'_{s_1}$.

The question I have is the following: Are such decompositions known as a general fact, or would I have to go through the process of changing the basis back to the standard basis and mapping back to the KL basis after the fact? Of course the latter can be done by hand (or computer for higher n) but I was just curious if there was a better way to approach this that I've overlooked as a non-expert.

Thanks!