Corollary

The group Aut$_0(\mathfrak{a})$ is open and closed in Aut($\mathfrak{a}$) in the Zariski topology.

Prop. 6 shows that Aut$_0(\mathfrak{a})$ is closed. Let $\overline{k}$ be an algebraic closure of $k$. The group Aut($\mathfrak{a}\otimes \overline{k}$)$/$Aut$_0(\mathfrak{a}\otimes \overline{k})$ is finite (§5, no. 3, Cor. 1 of Prop. 5); a fortiori, the group Aut($\mathfrak{a}$)$/$Aut$_0(\mathfrak{a})$ is finite. Since the cosets of Aut($\mathfrak{a}$) in Aut($\mathfrak{a}$) are closed, it follows that Aut$_0(\mathfrak{a})$ is open in Aut($\mathfrak{a}$).