The following result is mentioned in a dissertation
$$\mathrm{C\ell}_{p+k,q}(\mathbb{R})\cong \mathrm{C\ell}_{p,q+k}(\mathbb{R}), \text{ , if } p-q\equiv 1 \text{(mod 4)}$$
However, since that dissertation has many typos, and this result is only used once, I am quite unsure of its validity. Do someone have a reference for it?
P.S. I have searched this book and found nothing.