So, my question is the following:
I would like to know the precise definition of the kernel $\ker\phi$ of a Lie algebra homomorphism $\phi: \mathfrak{g}\to\mathfrak{g}'$. As far as I know, the kernel of a homomorphism $f:A\to B$, in general, is the set of the elements of $A$ that are mapped to the identity element of $B$. I have seen in Kernel of a Lie Algebra Homomorphism a precise definition in the special case of Lie algebra homomorphisms, going as the following $$\ker\phi=\{X\in\mathfrak{g}:\hspace{0.3cm} \phi(X)=0\}.$$ However, I am troubled with the element $0$. Is $0$ supposed to mean the "identity" element of $\mathfrak{g}'$ seen as a vector space, i.e. the zero vector $0\in\mathfrak{g}'$?
Thank you for any help.