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](https://en.wikipedia.org/wiki/Partition_(number_theory))



```
RRRR

RRR
R

RR
RR

```

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



are all simply connected nilpotent lie groups matrix groups over $\mathbb{R}$?
A connected Lie group admits a faithful finite dimensional representation (i.e. is a matrix group) just when it is a semidirect product $S \rtimes R$ of a connected and simply connected solvable Lie group $S$ and a connected linearly reductive Lie group $R$. See Hilgert and Neeb, **Structure and Geometry of Lie Groups**, p. 595, Theorem 16.2.7. (This book is very well written and detailed.) In particular, since nilpotent Lie groups are solvable, simply connected nilpotent Lie groups are matrix groups.There is no classification of nilpotent Lie groups, or even of nilpotent Lie algebras. As Igor Rivin points out, there are nonlinear nilpotent Lie groups. 


