Sum of Killing vector fields is a Killing vector field

Question

Let $(M,g)$ be a Riemannian manifold. A smooth vector field $X$ is called a Killing vector field if the flow of $X$ acts by isometries, or, equivalently, if $L_X g = 0$. Now why is the sum of Killing vector fields a Killing vector field?

Answer 1

In the comments there are more than one good answers to the question. However, there is some other way to think of it.

Recall that $\mathcal{X}(M)$, the space of vector fields on $M$, is the Lie algebra of the Lie group $\mathrm{diff}(M)$, the group of automorphisms of $M$. The space $\mathcal{K}(M)\subset\mathcal{X}(M)$, which consists of the Killing vector fields, is the Lie algebra of the group of isometries on $M$. Hence, $\mathcal{K}(M)$ is a vector space.