Suppose that $\\\\sigma:\\\\calm\\\\to\\\\calm$ is an ultraweakly continuous surjective $*$-linear mapping and $d:\\\\calm\\\\to\\\\calm$ is an ultraweakly continuous $*$-$\\\\sigma$-derivation such that $d(I)$ is a central element of $\\\\calm$. We provide a Kadison--Sakai type theorem by proving that $\\\\calh$ can be decomposed into ${\\\\calk}\\\\oplus{\\\\call}$ and $d$ can be factored as the form $\\\\delta\\\\oplus 2Z\\\\tau$, where $\\\\delta:{\\\\calm}\\\\to\\\\calm$ is an inner $*$-$\\\\sigma_{\\\\calk}$-derivation, $Z$ is a central element, $2\\\\tau=2\\\\sigma_{\\\\call}$ is a $*$-homomorphism, and $\\\\sigma_{\\\\calk}$ and $\\\\sigma_{\\\\call}$ stand for compressions of $\\\\sigma$ to $\\\\calk$ and $\\\\call$, respectively.

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