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 $bz+cz+ay = az+bx+cy$. 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.