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Given $a,b \in \mathfrak{su(n)}$ which generate the full algebra, it is possible to write and $G \in SU(n)$ as:

$G = \exp(\alpha_1 a)\exp(\beta_1 b) \ldots \exp(\alpha_m a)\exp(\beta_m b)$

for some coefficients $\alpha_k, \beta_k$ to be determined.

How many such exponentials are required for a given $G$, and is there any algorithm known to find these coefficients?

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    $\begingroup$ There's a result saying that if $a,b$ generate the Lie algebra, then $\exp(ta)$ and $\exp(t'b)$ generate a dense subgroup for $t,t'$ small enough. It's not enough to conclude but it can help (the answer is certainly yes). Possibly $m=m(a,b)$ has to tend to infinity when $[a,b]$ gets worse (e.g., $a,b$ belong to the 1-sphere and tend to commuting elements). $\endgroup$
    – YCor
    Commented Jan 11, 2018 at 16:45

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