Let $\alpha,\beta$ denote a $n\times 1$-vectors and $A,B$ $n\times n$-matrices. I would like to write
$$\alpha^TAB\beta=X\alpha^TB\beta, \ \ \rm{or,} \ \ \alpha^TAB\beta=\alpha^TB\beta X$$ for some $X$ which depends only on $\alpha,\beta$ and $A$ but not on $B$.
If we can write $A=\gamma\alpha^T$ for some $n\times 1$-vector $\gamma$ then $\alpha^T\gamma\alpha^T B\beta=X\alpha^TB\beta$ and $X=\alpha^T\gamma$. However this is a very special case. Are there any other circumstances for which we can find such an $X$? I would like a more general result if possible.
Help is much appreciated :)