  Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion.
As an application of a recent result of ours on exponential functionals of
Brownian motion, we show in this paper that, for every fixed $t>0$, the process
given by \begin{align*}
  B_{s}-B_{t}-\Bigl|
  B_{t}+\max _{0\le u\le s}B_{u}-\max _{s\le u\le t}B_{u}
  \Bigr|
  +\Bigl|
  \max _{0\le u\le s}B_{u}-\max _{s\le u\le t}B_{u}
  \Bigr| ,\quad 0\le s\le t, \end{align*} is a Brownian motion. The path
transformation that describes the above process is proven to be an involution,
commute with time reversal, and preserve Pitman's transformation. A connection
with Pitman's $2M-X$ theorem is also discussed.
