Return to Answer

The formula $$ e^{u[X,Y]}=\lim\limits_{N\to\infty}\left(e^{\sqrt{u/N}X}e^{\sqrt{u/N}Y}e^{-\sqrt{u/N}X}e^{-\sqrt{u/N}Y}\right)^N \tag{$*$} $$ (Godement 2017, p. 70) shows that $e^{u[X,Y]}\in G$ always, and seems sufficiently “like Lie’s” to be what you want.

Note added: The origin of this formula was left unresolved before. Howe (1983) and Godement (originally 1982) point to von Neumann (1929, §II.3) who proves $e^{u[X,Y]}\in G$, but apparently not $(*)$. Formula $(*)$ is explicit in e.g. Goldstein (1970), Nelson (1969, p. 111), Goto (1969, p. 159), Hausner-Schwartz (1968, p. 78), Cohn (1957, p. 112), and somewhat implicit in Gluškov (1957, p. 137), Yamabe (1950, p. 14), Yosida (1936, Remark p. 470).

The formula $$ e^{u[X,Y]}=\lim\limits_{n\to\infty}\left(e^{\sqrt{u/n}\,X}e^{\sqrt{u/n}\,Y}e^{-\sqrt{u/n}\,X}e^{-\sqrt{u/n}\,Y}\right)^n \tag{$*$} $$ (Godement 2017, p. 70) shows that $e^{u[X,Y]}\in G$ always, and seems sufficiently “like” “Lie’s” to be what you want.

Note added: The origin of this formula was left unresolved before. Howe (1983) and Godement (originally 1982) point to von Neumann (1929, §II.3) who proves $e^{u[X,Y]}\in G$, but apparently not $(*)$. Formula $(*)$ is explicit in e.g. Chorin et al. (1978, p. 207), Goldstein (1970), Nelson (1969, p. 111), Goto (1969, p. 159), Hausner-Schwartz (1968, p. 78), Cohn (1957, p. 112), and somewhat implicit in Gluškov (1957, p. 137), Yamabe (1950, p. 14), Yosida (1936, Remark p. 470). Almost none of whom cite each other!

The formula $$ e^{u[X,Y]}=\lim\limits_{N\to\infty}\left(e^{\sqrt{u/N}X}e^{\sqrt{u/N}Y}e^{-\sqrt{u/N}X}e^{-\sqrt{u/N}Y}\right)^N \tag{$*$} $$ (Godement 2017, p. 70) shows that $e^{u[X,Y]}\in G$ always, and seems sufficiently “like Lie’s” to be what you want.

Note added: The origin of this formula was left unresolved before. Howe (1983) and Godement (originally 1982) point to von Neumann (1929, §II.3) who proves $e^{u[X,Y]}\in G$, but apparently not $(*)$. Formula $(*)$ is explicit in e.g. Goldstein (1970), Nelson (1969, p. 111), Goto (1969, p. 159), Hausner-Schwartz (1968, p. 78), and somewhat implicit in Gluškov (1957, p. 137), Yamabe (1950, p. 14), Yosida (1936, Remark p. 470).

The formula $$ e^{u[X,Y]}=\lim\limits_{N\to\infty}\left(e^{\sqrt{u/N}X}e^{\sqrt{u/N}Y}e^{-\sqrt{u/N}X}e^{-\sqrt{u/N}Y}\right)^N \tag{$*$} $$ (Godement 2017, p. 70) shows that $e^{u[X,Y]}\in G$ always, and seems sufficiently “like Lie’s” to be what you want.

Note added: The origin of this formula was left unresolved before. Howe (1983) and Godement (originally 1982) point to von Neumann (1929, §II.3) who proves $e^{u[X,Y]}\in G$, but apparently not $(*)$. Formula $(*)$ is explicit in e.g. Goldstein (1970), Nelson (1969, p. 111), Goto (1969, p. 159), Hausner-Schwartz (1968, p. 78), Cohn (1957, p. 112), and somewhat implicit in Gluškov (1957, p. 137), Yamabe (1950, p. 14), Yosida (1936, Remark p. 470).

The formula $$ e^{u[X,Y]}=\lim\limits_{N\to\infty}\left(e^{\sqrt{u/N}X}e^{\sqrt{u/N}Y}e^{-\sqrt{u/N}X}e^{-\sqrt{u/N}Y}\right)^N \tag{$*$} $$ (Godement 2017, p. 70) shows that $e^{u[X,Y]}\in G$ always, and seems sufficiently “like Lie’s” to be what you want.

Note added: The origin of this formula was left unresolved before. Howe (1983) and Godement (originally 1982) point to von Neumann (1929, §II.3) who proves $e^{u[X,Y]}\in G$, but apparently not $(*)$. Formula $(*)$ is explicit in e.g. Goldstein (1970), Nelson (1969, p. 111), Goto (1969, p. 159), Hausner-Schwartz (1968, p. 78), and somewhat implicit in Gluškov (1957, p. 137), Yamabe (1950, p. 14), Yosida (1936, Remark p. 470).

The formula $$ e^{u[X,Y]}=\lim\limits_{N\to\infty}\left(e^{\sqrt{u/N}X}e^{\sqrt{u/N}Y}e^{-\sqrt{u/N}X}e^{-\sqrt{u/N}Y}\right)^N \tag{$*$} $$ (Godement 2017, p. 70) shows that $e^{u[X,Y]}\in G$ always, and seems sufficiently “like Lie’s” to be what you want.

Note added: The origin of this formula was left unresolved before. Howe (1983) and Godement (originally 1982) point to von Neumann (1929, §II.3) who proves $e^{u[X,Y]}\in G$, but apparently not $(*)$. Formula $(*)$ is explicit in e.g. Goldstein (1970), Nelson (1969, p. 111), Goto (1969, p. 159), Hausner-Schwartz (1968, p. 78), and somewhat implicit in Gluškov (1957, p. 137), Yamabe (1950, p. 14), Yosida (1936, Remark p. 470).