Let $(X,d)$ be a compact metric space and let $K$ be a nonempty compact subset of $X$. Let $\alpha \in (0, 1]$ and let ${\rm Lip}(X,K,d^ \alpha)$ denote the Banach algebra of all continuous complex-valued functions $f$ on $X$ for which $p_{(K,d^\alpha)}(f)=\sup\{\frac{|f(x)-f(y)|}{d^\alpha(x,y)} : x,y\in K , x\neq y\}<\infty$ when equipped the algebra norm $||f||_{{\rm Lip}(X, K, d^ {\alpha})}= ||f||_X+ p_{(K,d^{\alpha})}(f)$, where $||f||_X=\sup\{|f(x)|:~x\in X \}$. We denote by ${\rm lip}(X,K,d^ \alpha )$ the closed subalgebra of ${\rm Lip}(X,K,d^ \alpha)$ consisting of all $f\in {\rm Lip}(X,K,d^ \alpha)$ for which $\frac{|f(x)-f(y)|}{d^\alpha(x,y)}\to 0$ as $d(x, y)\to 0$ with $x, y \in K$. In this paper we show that every proper closed ideal of $({\rm lip}(X,K,d^\alpha),\|\cdot\|_{Lip(X,K,d^\alpha)})$ is the intersection of all maximal ideals containing it. We also prove that every continuous point derivation of ${\rm lip}(X,K,d^\alpha)$ is zero. Next we show that ${\rm lip}(X,K,d^\alpha)$ is weakly amenable if $\alpha \in (0, \frac{1} {2})$. We also prove that ${\rm lip}(\Bbb {T},K,d^{\frac{1}{2}})$ is weakly amenable where $\Bbb {T}=\{z\in\Bbb C : |z|=1\}$, $d$ is the Euclidean metric on $\Bbb T$ and $K$ is a nonempty compact set in $(\Bbb T,d)$.
