Quaternion-valued d’Alembert functions on semigroups

Let S be a semigroup with an involution \(\tau \) $\tau$ (meaning that it is an involutive anti-automorphism of S)and let \(\mathbb {H}\) $\mathbb{H}$ be the skew field of quaternions. Our main goal is to solve d’Alembert’s quaternion functional equation \(\begin{aligned} L(xy)+L(x\tau (y))=2L(x)L(y),\,\,\,x,y\in S, \end{aligned}\) $$\begin{array}{c}\hfill L(xy)+L(x\tau (y))=2L(x)L(y),x,y\in S,\end{array}$$ where \(L:S\rightarrow \mathbb {H}\) $L:S\to \mathbb{H}$ is the unknown function.