Isotropy indices of Pfister multiples in characteristic two

Let $F$ be a field of characteristic $2$, $\pi$ be an $n$-fold bilinear Pfister form over $F$ and $\phi$ an arbitrary quadratic form over $F$. In this note, we investigate Witt index, defect, total isotropy index and higher isotropy indices of $\phi$ and $\pi \otimes \phi$ and prove relations among the indices of these two forms over certain field extensions.