# Kernel of adjoint of Lie algebra

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra.

The adjoint representation of the Lie algebra $\mathfrak{g}$ is defined as:

$$\text{ad: } \mathfrak{g} \rightarrow \text{End}(\mathfrak{g}), X \mapsto [X,\cdot]$$

Now, it holds true that

$$\text{ker ad} = \mathfrak{z}(\mathfrak{g}) = \{X \in \mathfrak{g} : [X,Y] = 0 \quad\forall\; Y \in \mathfrak{g}\}.$$

On the other hand, the definition of the kernel of this homomorphism is (at least in my mind)

$$\text{ker ad} = \{ X \in \mathfrak{g} : [X,\cdot] = \text{id}, \text{ i.e. } [X,Y] = Y \quad \forall \;Y \in \mathfrak{g} \},$$

since the group identity in the endomorphism group is the identity-map.

Evidently, the two sets are not the same, but where is my mistake?

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This is a homomorphism of Lie algebras, not groups. The kernel is as a linear map. i.e. the preimage of zero. – KotelKanim Aug 8 '12 at 17:17