  We give an $SL_3$ analogue of the triangular decomposition of the Kauffman
bracket stated skein algebras described by Le. To any punctured bordered
surface, we associate an $SL_3$ stated skein algebra which contains the $SL_3$
skein algebra of closed webs. These algebras admit natural algebra morphisms
associated to the splitting of surfaces along ideal arcs. We give an explicit
basis for the $SL_3$ stated skein algebra and show that the splitting morphisms
are injective and describe their images. By splitting a surface along the edges
of an ideal triangulation, we see that the $SL_3$ stated skein algebra of any
ideal triangulable surface embeds into a tensor product of stated skein
algebras of triangles.
