  We address the problem of finite products that attain the joint spectral
radius of a finite number of square matrices. Up to date the problem of
existence of "forbidden products" remained open. We prove that the product
$AABABABB$ (together with its circular shifts and their mirror images) never
delivers the strict maximum to the joint spectral radius if we restrict
consideration to pairs $\{A,B\}$ of real $2\by 2$ matrices. Under this
restriction circular shifts and their mirror images constitute the class of
isospectral products and hence they all have the same spectral radius for any
pair $\{A,B\}$ of $2\by 2$ matrices, even complex. For pairs of complex
matrices we have numerical evidence that $AABABABB$ is still a fobidden
product. A couple of binary words that encode products from this isospectral
class also happen to be the shortest forbidden patterns in the parametric
family of double rotations.
