Controller/Observer Canonical Forms
For example, have you considered the following factorisation, with each term expressed in controller canonical form?

$$\small \frac{1}{(s+1)}\:\times \small \frac{1}{(s+1)}$$

This is neither controllable nor observable. Check it out with \$\small\left[B\:\:AB\right]\$ and \$\small\left[C\:\:CA\right]^{T}\$.