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Given a connected lie group $G$, since a neighbourhood of the origin generates all of $G$, we have that every $g$ in $G$ can be expressed as a finite product of elements of the form $e^X$, for $X$ in the lie algebra $\mathfrak{g}$ of $G$.

Let us define $f(G)$ to be the minimal $n\in \mathbb{N}\cup \{\infty\}$ such that any element of $G$ can be expressed as a product of $n$ elements of the form $e^X$ for $X$ in $\mathfrak{g}$.

It's a theorem that for compact $G$, we have $f(G)=1$, which is another way of saying $exp$ is surjective, and we can check that $exp$ is surjective for the the group of upper(or lower) triangular matrices with $1$s on the diagonal, so by factoring any matrix in $GL_n(\mathbb{C})$ as $UDU'$ with $U,U'$ of this form, and $D$ diagonal, we also see that $f(GL_n(\mathbb{C}))\leq 3$.

One can do similar things with $SL_n$, and the group of upper triangular matrices, and get that in each case, $f(G)$ is finite.

Is $f(G)$ always finite? What about for linear lie groups?

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