  Let $p$ be a prime such that the multiplicative order $m$ of $2$ modulo $p$
is even. We prove that the integral group ring $\mathbb{Z}[Q_8 \times C_p]$ has
the multiplicative Jordan decomposition property when $m$ is congruent to $2$
modulo $4$. There are infinitely many such primes and these primes include the
case $p \equiv 3 \pmod{4}$. We also prove that $\mathbb{Z}[Q_8 \times C_5]$ has
the multiplicative Jordan decomposition property in a new way.
