Consider the set of $n\times n$ upper-triangular matrices $\mathfrak b$. Let $B$ be the set of invertible $n\times n$ upper-triangular matrices. Consider the following action:
$$\begin{array}{rll} (B\times B)\times \mathfrak b&\longrightarrow \mathfrak b\\ (b_1,b_2)\times b&\longmapsto &b_1bb_2^{-1} \end{array} $$
(1) Give a complete classification of orbits of this action.
(2) For each orbit $\mathfrak O$, consider the following action $$\begin{array}{rll} B\times \mathfrak O&\longrightarrow \mathfrak O\\ b_1\times b&\longmapsto &b_1bb_1^{-1} \end{array} $$ give a complete classification of orbits for this actions for each $\mathfrak O$.