Given the following Lie transformation:
$$ \exp(\lbrace H, \cdot \rbrace):=\sum_{n=0}^{\infty} \frac{(\lbrace H, \cdot \rbrace)^n}{n!} $$ and apply it to a Poisson Bracket $\lbrace g_1, g_2 \rbrace$. I would like to show that $$ \exp(\lbrace H, \lbrace g_1, g_2 \rbrace\rbrace)=\sum_{n=0}^{\infty} \frac{(\lbrace H, \lbrace g_1, g_2 \rbrace \rbrace)^n}{n!}=\lbrace \sum_{n=0}^{\infty} \frac{(\lbrace H, g_1\rbrace)^n}{n!}, \sum_{n=0}^{\infty} \frac{(\lbrace H, g_2\rbrace)^n}{n!}\rbrace=\lbrace\exp(\lbrace H, g_1\rbrace),\exp(\lbrace H, g_2\rbrace)\rbrace $$
I am sure this property holds, since I found this here (page 25): http://www.aps.anl.gov/Science/Publications/lsnotes/content/files/APS_1418211.pdf
but I have difficulties in proving this in an elegant way. Could someone give me a good reference?