Let
F
F
be a finite type surface and
ζ
\zeta
a complex root of unity. The Kauffman bracket skein algebra
K
ζ
(
F
)
K_\zeta (F)
is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the rank and trace of
K
ζ
(
F
)
K_\zeta (F)
over its center, and we extend a theorem of the first and second authors in [Math. Z. 289 (2018), pp. 889–920] which says the skein algebra has a splitting coming from two pants decompositions of
F
F
.