Q:

Simple (?) but frustrating: transforming a $2\times 2$ covariance matrix

Suppose I have a matrix
$$
\underset{2\times 2}{V}=\begin{pmatrix}\sigma_x^2 & \sigma_{xy} \\ \sigma_{xy} & \sigma_y^2\end{pmatrix}
$$
where the RHS quantities above correspond in obvious ways to the variances and covariance of 2 random variables $X$ and $Y$. Let $u$ and $v$ be the 1st and 2nd columns of $V$ respectively (both $u$ and $v$ are $2\times 1$). I'm interested in the following quantity:
$$
\underset{4\times 4}R=\begin{pmatrix}
uu' & vu' \\
uv' & vv'
\end{pmatrix}
$$
and how I can write it in terms of $V$, but not explicitly in terms of $u$ and $v$. (In my real application, there is one $X$ still but the number of $Y$'s is more than $1$.)
So far, I have tried:
$$
\operatorname{vec}(V)\cdot[\operatorname{vec}(V)]'\quad\text{and}\quad V\otimes V\tag{$*$}
$$
where $\otimes$ represents the Kronecker product. Neither of these produces $R$ when $V=\begin{pmatrix}1 & 0.5 \\ 0.5 & 1\end{pmatrix}$.

A:

EDITED: Your matrix is
$$ \pmatrix{V & 0\cr 0 & V}\pmatrix{1 & 0 & 0 & 0\cr 0 & 0 & 1 & 0\cr
0 & 1 & 0 & 0\cr 0 & 0 & 0 & 1} \pmatrix{V & 0\cr 0 & V} = (I_2 \otimes V) \pmatrix{1 & 0 & 0 & 0\cr 0 & 0 & 1 & 0\cr
0 & 1 & 0 & 0\cr 0 & 0 & 0 & 1} (I_2 \otimes V) $$

