An example of a unipotent matrix which is NOT upper triangular Define::=  $I_n$ -- the $n \times n$ identity matrix.
Let $A$ be an $n \times n$ real matrix.
Define::=  Nilpotent matrix -- an $n \times n$ real matrix $X$ such that $X^n = $ the zero matrix for some $n$ in the positive integers.
Define::= unipotent matrix $U$  -- $A - I$ which is nilpotent.
I'd like an example of a unipotent matrix which is not upper triangular

$$A=\left[ \begin{array}{ccc}1&1&0\\ 0&1&0\\ 0&1&1\end{array}\right]$$is unipotent because $(A-I)^2=0$ and $(A-I)$ is nilpotent where $I$ is the identity (or one can also argue that the characteristic polynomial of $A$ is $(\lambda-1)^3$) whereas A is neither upper nor lower triangular.
