I just encountered a very curious relation in an algebra. A bit simplified, I am working in a (particular) non-commutative algebra, with some relations.

One particular relation is the following:
For (some particular choices of) 
elements $a,b,c,x,y,z$ in the algebra,
the relation is of the form $bz+cx+ay = az+bx+cy$. 
This can be stated as a vanishing 3x3-determinant,

$$
\begin{vmatrix}
1 & 1 & 1 \\
a & b & c \\
x & y & z
\end{vmatrix}=0.
$$ 

We can think of $a,...,z$ as words, and multiplication is simply concatenation. **Has this type of relation been observed/used in some algebras before**? Does it fit into some larger picture?

In my concrete case, this relation does not hold in general, but only for certain particular combinations of $a,b,c,x,y,z$, so examples where there are some restrictions on when the relation applies are also interesting to me.