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I am pretty sure that the first use of Fraktur by Sophus Lie occurs in 1869, which is before he invented Lie groups or Lie algebras. It appears in his paper Repraesentation der Imaginaeren der Plangeometrie, in the first volume of his collected works, to represent the plane. I assume that it was standard practice in German mathematics to use German script letters, because they were used in number theory by Dirichlet and others before 1869. Lie groups slowly evolved in the mid to late 1870's, entering their final form in the 1880s.