# Complexification of real inner product spaces and how the inner product extends to a complex space

I am learning about complexification of a real vector space, say $X$, and I understand how $[\mathbf x_1 , \mathbf x_2]$ is defined as $\mathbf x_1 + i\mathbf x_2$, and that $i[\mathbf x_1 , \mathbf x_2] = [-\mathbf x_2 , \mathbf x_1]$, but am having some trouble understanding how exactly the inner product in $X$ extends to $X_C$ (the complex space constructed from a real space X)

I made up some numbers to work with the formula $([\mathbf x_1 , \mathbf x_2] , [\mathbf y_1 , \mathbf y_2])_{X_C} = (\mathbf x_1, \mathbf y_1)_X + i(\mathbf x_2, \mathbf y_2)_X$given in the notes I'm referring to, namely as follows:

$\mathbf x_1 = \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix}$, $\mathbf x_2 = \begin{pmatrix} 3 \\ 4 \\ \end{pmatrix}$, $\mathbf y_1 = \begin{pmatrix} 5 \\ 6 \\ \end{pmatrix}$ and $\mathbf y_2 = \begin{pmatrix} 1 \\ 4 \\ \end{pmatrix}$

$[\mathbf x_1 , \mathbf x_2] = \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} + i\begin{pmatrix} 3 \\ 4 \\ \end{pmatrix} = \begin{pmatrix} 1 + 3i \\ 2 +4i \\ \end{pmatrix}$

$[\mathbf y_1 , \mathbf y_2] = \begin{pmatrix} 5 \\ 6 \\ \end{pmatrix} + i\begin{pmatrix} 1 \\ 4 \\\end{pmatrix} = \begin{pmatrix} 5 + i \\ 6 + 4i \\ \end{pmatrix}$

Using the definition of the inner product $(\mathbf u, \mathbf v) = \mathbf v^* \mathbf u$, I get $([\mathbf x_1 , \mathbf x_2] , [\mathbf y_1 , \mathbf y_2])_{X_C} = (5-i)(1+3i) + (6-4i)(2+4i) = 36 + 30i$ ,

but using $([\mathbf x_1 , \mathbf x_2] , [\mathbf y_1 , \mathbf y_2])_{X_C} = (\mathbf x_1, \mathbf y_1)_X + i(\mathbf x_2, \mathbf y_2)_X$, I get

$([\mathbf x_1 , \mathbf x_2] , [\mathbf y_1 , \mathbf y_2])_{X_C} = (\begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} , \begin{pmatrix} 5 \\ 6 \\ \end{pmatrix}) + i(\begin{pmatrix} 3 \\ 4 \\ \end{pmatrix} , \begin{pmatrix} 1 \\ 4 \\ \end{pmatrix}) = 5 + 12 + i(3 + 16) = 17 + 19i$.

What went wrong with my interpretation of the formula , $([\mathbf x_1 , \mathbf x_2] , [\mathbf y_1 , \mathbf y_2])_{X_C} = (\mathbf x_1, \mathbf y_1)_X + i(\mathbf x_2, \mathbf y_2)_X$ ? Shouldn't both computations be mathematically sound and give the same result? Thanks in advance for any help!