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I have this matrix:

$$A = left(
beginarraycccc
0 & 0 & 0 & -1 \
1 & 0 & 0 & -1 \
0 & 1 & 0 & -1 \
0 & 0 & 1 & -1 \
endarray
right)$$

and I would like to show that it can be block diagonalised into :

$$ B = left(
beginarraycccc
cos 2pi/5 & -sin 2pi/5 & 0 & 0 \
sin 2pi/5 & cos 2pi/5 & 0 & 0 \
0 & 0 & cos 4pi/5 & -sin 4pi/5 \
0 & 0 & sin 4pi/5 & cos 4pi/5\
endarray
right) = left(
beginarraycccc
frac14 left(-1+sqrt5right) & -sqrtfrac58+fracsqrt58 & 0 & 0 \
sqrtfrac58+fracsqrt58 & frac14 left(-1+sqrt5right) & 0 & 0 \
0 & 0 & frac14 left(-1-sqrt5right) & -sqrtfrac58-fracsqrt58 \
0 & 0 & sqrtfrac58-fracsqrt58 & frac14 left(-1-sqrt5right) \
endarray
right)$$

What is the general procedure?

When you diagonalise a matrix over the complex numbers, you start by finding its eigenvalues, which may be complex. If the matrix is diagonalisable, then we can expect a basis of eigenvectors corresponding to these eigenvalues. If we compute the transformation $x mapsto Ax$ in terms of this basis of eigenvectors, we get a complex diagonal matrix, similar to $A$.

To block-diagonalise $A$ over the reals, again find all the complex eigenvalues and a basis of eigenvectors. Suppose $alpha + i beta$ is a non-real eigenvalue, with corresponding eigenvector $v + i w$, where $v, w$ are vectors with real components. One can easily verify that $alpha - i beta$ is another eigenvalue with eigenvector $v - i w$. We can form a basis of eigenvectors such that, if $v + iw$ is in the basis (where $v$ and $w$ are real vectors), then so is $v - iw$.

Now, form a real basis like so: replace the conjugate eigenvector pairs $v pm i w$ with the vectors $v, w$. If $v + iw$ has an eigenvalue $alpha + i beta$, then
beginalign*
Av &= A left(frac(v + iw) + (v - iw)2right) \
&= frac(alpha + ibeta)(v + iw) + (alpha - ibeta)(v - iw)2 \
&= fracalpha v + i beta v + i alpha w - beta w + alpha v - i beta v - i alpha w - beta w2 \
&= alpha v - beta w.
endalign*
Similar calculation reveals
$$Aw = beta v + alpha w.$$
So, when computing the matrix for $x mapsto Ax$ with this basis, you'll find a block diagonal form, where each block is either a $1 times 1$ block containing a real eigenvalue, or a $2 times 2$ block of the form
$$beginpmatrix alpha & beta \ -beta & alpha endpmatrix,$$
where $alpha pm beta i$ is a non-real eigenvalue of $A$.

it is already a companion matrix (both that form and its transpose are used, depends on circumstances). The characteristic polynomial is $$x^4 + x^3 + x^2 + x + 1 = fracx^5-1x-1$$
With four distinct eigenvalues (complex) it diagonalizes. Next, you need to figure out how to take a specific diagonal matrix, with complex $alpha$ such that $|alpha| = 1,$
$$
left(
beginarrayrr
alpha & 0 \
0 & baralpha
endarray
right)
$$
and send it back to one of your two by two blocks with sine and cosine.

To reverse the job: with some real angle $theta,$ exactly how do you diagonalize
$$
left(
beginarrayrr
cos theta & -sin theta \
sin theta & cos theta
endarray
right) ; ; ? ;
$$