  Recently, Ikoma (2022) considered optimal constants and extremisers for the
$2$-dimensional Dirac equation using the spherical harmonics decomposition.
Though its argument is valid in any dimensions $d \geq 2$, the case $d \geq 3$
remains open since it leads us to too complicated calculation: determining all
eigenvalues and eigenvectors of infinite dimensional matrices. In this paper,
we give optimal constants and extremisers of smoothing estimates for the
$3$-dimensional Dirac equation. In order to prove this, we construct a certain
orthonormal basis of spherical harmonics. With respect to this basis, infinite
dimensional matrices actually become block diagonal and so that eigenvalues and
eigenvectors can be easily found. As applications, we obtain the equivalence of
the smoothing estimate for the Schr\"{o}dinger equation and the Dirac equation,
and improve a result by Ben-Artzi and Umeda (2021).
