The $\mathbb{C}$-motivic wedge subalgebra

We describe some regular behavior in the motivic wedge, which is a subalgebra of the cohomology Ext${}_{\mathbf{A}}({\mathbb{M}}_2,{\mathbb{M}}_2)$ of the $\mathbb{C}$-motivic Steenrod algebra. The key tool is to compare motivic computations to classical computations, to Ext${}_{\mathbf{A}(2)}({\mathbb{M}}_2,{\mathbb{M}}_2)$, or to $h_1$-localization of Ext${}_{\mathbf{A}}({\mathbb{M}}_2,{\mathbb{M}}_2)$. We also give a conjecture on the behavior of the family $e_0^t{g}^k$ in Ext${}_{\mathbf{A}}({\mathbb{M}}_2,{\mathbb{M}}_2)$ which raises naturally from the study of the motivic wedge.