I am studying Helgason's Differential Geometry and Symmetric Spaces, trying to understand real forms of...

I am studying Helgason's Differential Geometry and Symmetric Spaces, trying to understand real forms of Lie Algebras.
My problem is related to Lemma 6.1 (1st edition):
Let $K_0$ be the Killing form of a Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, and $K$ the Killing form of the complexification Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$. Then
$K_0(X,Y)=K(X,Y)\mathrm{\forall }X,Y\in \mathfrak{g}$

My problem:
Understanding the previous equation as $K(X,Y)\equiv K(X+i0,Y+i0)$ I would get twice the result stated in the book. This is due to the following facts.
1. ${\text{ad}}_{\mathbb{C}}(X)\in {\text{End}}_{\mathbb{C}}({\mathfrak{g}}_{\mathbb{C}})$, acting as
${\text{ad}}_{\mathbb{C}}(X)[A+iB]=[X+i0,A+iB{]}_{\mathbb{C}}=[X,A]+i[X,B]$
therefore I can write it as a direct sum of the real adjoint map
${\text{ad}}_{\mathbb{C}}(X)=\text{ad}(X)\oplus \text{ad}(X)$
2. $\text{Trace}[{\text{ad}}_{\mathbb{C}}(X)\circ {\text{ad}}_{\mathbb{C}}(Y)]=\text{Trace}[\text{ad}(X)\circ \text{ad}(Y)\oplus \text{ad}(X)\circ \text{ad}(Y)]=2\text{Trace}[\text{ad}(X)\circ \text{ad}(Y)]$

Background:
1. Helgason's definition of complexification: ${\mathfrak{g}}_{\mathbb{C}}=\mathfrak{g}\times \mathfrak{g}\simeq \mathfrak{g}\oplus \mathfrak{g}$ with the complex structure
$J:(X,Y)\equiv X+iY\mapsto (-Y,X)\equiv -Y+iX$
extending the Lie bracket by $\mathbb{C}$-linearity:
$[X+iY,Z+iT{]}_{\mathbb{C}}=[X,Z]-[Y,T]+i[X,T]+i[Y,Z]$
2. Killing form of any Lie algebra over an arbitrary field $\mathbb{K}$:
$B(X,Y)=\text{Trace}[\text{ad}(X)\circ \text{ad}(Y)]$
where $\text{ad}(X)$ is the $\mathbb{K}$-linear map $[X,\cdot ]$.