For a semisimple Lie algebra $\mathfrak{g}$ and a marked surface $\Sigma$, the $\mathfrak{g}$-skein algebra is a noncommutative algebra generated by $\mathfrak{g}$-webs on $\Sigma$ subject to the $\mathfrak{g}$-skein relations. A $\mathfrak{g}$-web is a trivalent graph embedded in $\Sigma\times [0,1]$, with edges colored by fundamental representations of $\mathfrak{g}$.
In this talk, we focus on the case $\mathfrak{g}=\mathfrak{sp}_4$ and introduce a filtration of the $\mathfrak{g}$-skein algebra associated with an ideal triangulation. We also discuss a relationship between its associated graded algebra and an $\mathfrak{sp}_4$-lamination on $\Sigma$. This talk is based on joint work with T. Ishibashi and Z. Sun.