Quadratic form reduction on the $n$-dimensional complex space $\mathbb{C}^n$ How to reduce the quadratic form
$$\sum_{1 \leq k < l \leq n}(k + il)x_kx_l$$
to the canonical form on the $n$-dimensional complex space $\mathbb{C}^n$?
I started with its symmetric matrix
\begin{align*}
S = \frac{1}{2}\begin{pmatrix}
0 & 1 + 2i & 1 + 3i & \cdots & 1 + ni \\
1 + 2i & 0 & 2 + 3i & \cdots & 2 + ni \\
1 + 3i & 2 + 3i & 0 & \cdots & 3 + n i \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 + ni & 2 + ni & 3 + ni & \cdots & 0
\end{pmatrix},
\end{align*}
and then tried to reduce it to the diagonal matrix following the algorithmic recipe, i.e., tried reducing it by eliminating the off diagonal elements row/column after row/column: the first step is to clear the first two rows/columns by calculating $P^TSP$, where
\begin{align*}
P = \begin{pmatrix}
1 & -1 & 0 & \cdots & 0 \\
1 &  1 & 0 & \cdots & 0 \\
0 &  0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 &  0 & 0 & \cdots & 1
\end{pmatrix}
\begin{pmatrix}
1 &  0 & -\frac{1 + 3i}{1 + 2i} & \cdots & -\frac{1 + ni}{1 + 2i} \\
0 &  1 & \frac{1}{2(1 + 2i)} & \cdots & \frac{1}{2(1 + 2i)} \\
0 &  0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 &  0 & 0 & \cdots & 1
\end{pmatrix}.
\end{align*}
The resultant matrix $P^TSP := (s_{ij})$ has already quite complicated entries:
\begin{align*}
& s_{kk} = \frac{k^2 - 2 - 3ki}{1 + 2i}, \; k = 3, 4, \ldots, n; \\
& s_{kl} = \frac{k - 2l + 2kl - 4 - li}{2(1 + 2i)}, \; l > k,
\end{align*}
making the subsequent calculation notoriously nasty, so I am still inconclusive.
Is my attempt on the right track to solve this problem? I think there probably should be a better and cleaner way.

This might be helpful
If we define
$$
\Phi_1:=\sum_{1\leq k<l\leq n}(k+il)x_kx_l
$$
and
$$
\epsilon_{k,l}=\left\{\begin{array}{cc}
1\textrm{ , if }k<l\\
i\textrm{ , if }k>l\\
0\textrm{ , else }
\end{array}\right\}\textrm{, }i=\sqrt{-1}
$$
then
$$
\Phi_1=\sum^{n}_{k,l=1}\epsilon_{kl}kx_kx_l=\frac{1+i}{1-i}\sum_{k<l}(k+l)x_kx_l-\frac{2i}{1-i}\sum^{n}_{k<l}kx_kx_l
$$
