The Terwilliger algebras of association schemes over an arbitrary field $\mathbb{F}$ were called the Terwilliger $\mathbb{F}$-algebras of association schemes in [10]. In [7], He and Jiang studied the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. In this paper, we continue studying the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We get all block idempotents of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes. We get the $\mathbb{F}$-dimensions, the centers, the Jacobson radicals of the block algebras of the Terwilliger $\mathbb{F}$-algebras of factorial association schemes.