The $\alpha$-Family in the $K(2)$-Local Sphere at the Prime $2$

In this note, we compute the image of the $\alpha$-family in the homotopy of the $K(2)$-local sphere at the prime $p=2$ by locating its image in the algebraic duality spectral sequence. This is a stepping stone for the computation of the homotopy groups of the $K(2)$-local sphere at the prime $2$ using the duality spectral sequences.