Our aim is to present new expressions for the Drazin inverse of anti-triangular block matrices under some circumstances. Applying the established new formulae for anti-triangular block matrices, we derive explicit representations for the Drazin inverse of a $2\times 2$ complex block matrix under corresponding assumptions. We extend several well known results in the literature in this way.

Let $\mathfrak{L}$ and $A$ be Lie triple systems, and let ${\theta }_A$ be a representation of $\mathfrak{L}$ on $A.$ We first construct the third-order cohomology classes by derivations of $A$ and $\mathfrak{L},$ then obtain a Lie algebra $G_{{\theta }_A}$ with a representation $\mathrm{\Phi }$ on $H^3(\mathfrak{L},A),$ where ${\theta }_A$ is given by an abelian extension

$$0⟶A⟶\tilde{\mathfrak{L}}\stackrel{\pi }{\to }\mathfrak{L}⟶0.$$

We study obstruction classes for extensibility of derivations of $A$ and $\mathfrak{L}$ to those of $\tilde{\mathfrak{L}}.$ An application of $\mathrm{\Phi }$ is discussed.