  Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion
and denote by $A_{t},\,t\ge 0$, the quadratic variation of $e^{B_{t}},\,t\ge
0$. The celebrated Bougerol's identity in law (1983) asserts that, if $\beta
=\{ \beta _{t}\} _{t\ge 0}$ is another Brownian motion independent of $B$, then
$\beta _{A_{t}}$ has the same law as $\sinh B_{t}$ for every fixed $t>0$.
Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the
identity involving as the second coordinates the local times of $B$ and $\beta
$ at level zero. In this paper, we present a generalization of their extension
in a situation that the levels of those local times are not restricted to zero.
Our argument provides a short elementary proof of the original extension and
sheds new light on that subtle identity.
