Brownian motion $g_t$ on a compact Lie group satisfies the stochastic differential equation

$$dg_t = dB_t \circ g_t$$

where $B_t$ is Brownian motion on the Lie algebra and $\circ$ denotes differential in the sense of Stratonovich. We take $g_0 = 1$. (We assume the Lie group and its Lie algebra are embedded into a group of matrices and we suppressed the matrix multiplication indices in the above.)

What's the analogous equation for a Brownian bridge, i.e. condition at time $T$ so that $g_T = 1$?