Mathworld just says that the lower central series terminates: $\mathfrak{g}_1= [ \mathfrak{g}, \mathfrak{g}]$, $\mathfrak{g}_2= [ \mathfrak{g}, \mathfrak{g}_1]$ and $\mathfrak{g}_n= [ \mathfrak{g}, \mathfrak{g}_{n-1}]$ the best example I could think of is something like:

$$ \left( \begin{array}{cccc} 1 & \mathbb{R} & \mathbb{R} & \mathbb{R} \\ 0 & 1 & 0 & \mathbb{R} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$

Another question on here says all simply connected nilpotent lie groups are matrix groups. It was phrased in a difficult way, so I am asking it again.

For example, another arrangement of $\mathbb{R}$'s that I can think of (at least in a $4 \times 4$ matrix) could be:

$$ \left( \begin{array}{cccc} 1 & 0 & \mathbb{R} & \mathbb{R} \\ 0 & 1 & \mathbb{R} & \mathbb{R} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$

and I think there is only one more in the $5 \times 5$ case:

$$ \left( \begin{array}{ccccc} 1 & \mathbb{R} & \mathbb{R} & \mathbb{R} & \mathbb{R} \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right)$$

So in other words, I am claiming (almost surely false) an identification with nilpotent lie algebras and Ferrers boards (integer partitions

```
RRRR
RRR
R
RR
RR
```

This is surely wrong. Is there a classification of nilpotent Lie groups? Are all nilpotent Lie Groups matix groups?