I am not able give an example for the following problem on [simultaneous triangularization](http://en.wikipedia.org/wiki/Triangular_matrix#Simultaneous_triangularisability). So, I thought I will post it here. 


Give an example of three linear transformations $A,B$ and $C,$ such that the pairs $\lbrace A,B\rbrace$, $\lbrace B,C\rbrace$ and $\lbrace A,C\rbrace$ are simultaneously triangularizable, but the triplet $\lbrace A,B,C\rbrace$ is not simultaneously triangularizable. 


Thank you. 


ADDED LATER: For my need, I am looking for an example of linear transformations acting on vector spaces over $\mathbb{C}.$ 



Example for pairwise triangularizable but not all three. 
Here is a simple example:
$$ A = \begin{pmatrix}
 0 & 1 & \\
 & 0 & \\
 & & 0
 \end{pmatrix}, 
 \quad
 B = \begin{pmatrix}
 0 & & \\
 & 0 & 1 \\
 & & 0
 \end{pmatrix},
 \quad
 C = \begin{pmatrix}
 0 & & \\
 & 0 & \\
 1 & & 0
 \end{pmatrix} .
$$
Every pair can be triangularized (by a permutation matrix, by the way), but $A$, $B$ and $C$ have no common eigenvector, and so these three matrices can't be simultaneously triangularized. 


