The most powerful scale and location invariant test of normality against the double exponential alternative is derived by the technique of integrating with respect to the scale and location transformation group. The resultant test is asymptotically equivalent to the likelihood ratio test of this hypothesis and to Geary's test (i.e. mean deviation over standard deviation) for all three test statistics are shown to have the same asymptotic normal distribution when the sampling is from a symmetric, absolutely continuous distribution, whose density is continuous in the neighborhood of its median and whose fourth moment exists.
"The Most Powerful Scale and Location Invariant Test of the Normal Versus the Double Exponential." Ann. Statist. 1 (1) 170 - 174, January, 1973. https://doi.org/10.1214/aos/1193342395