  The ground-state mass excess of the $T_{z}=-2$ drip-line nucleus $^{22}$Al is
measured for the first time to be $18103(10)$ keV using the newly-developed
B$\rho$-defined isochronous mass spectrometry method at the cooler storage ring
in Lanzhou. The new mass excess value allowed us to determine the excitation
energies of the two low-lying $1^+$ states in $^{22}$Al with significantly
reduced uncertainties of 51 keV. Comparing to the analogue states in its mirror
nucleus $^{22}$F, the mirror energy differences of the two $1^+$ states in the
$^{22}$Al-$^{22}$F mirror pair are determined to be $-625(51)$ keV and
$-330(51)$ keV, respectively. The excitation energies and the mirror energy
differences are used to test the state-of-the-art \textit{ab initio}
valence-space in-medium similarity renormalization group calculations with four
sets of interactions derived from the chiral effective field theory. The
mechanism leading to the large mirror energy differences is investigated and
attributed to the occupation of the $\pi s_{1/2}$ orbital.
