How do I determine if a given pure two-qubit state is separable?

I’m trying to self-study some topics about quantum computing and I came across a topic of state separability. Talking about that, I wanted to determine separability on the following state (from Qiskit Textbook):

$$
frac{1}{sqrt{2}}left|00right> + frac{i}{sqrt{2}}left|01right>
$$

I know, that it is separable, as it can be rewritten as

$$
left| 0right> otimes frac{left|0 right> + ileft| 1right>}{sqrt{2}},
$$

but what’s the way to determine if the state is separable or entangled without guessing?

It’s stated in Is there an algorithm for determining if a given vector is separable or entangled? , there are ways to do this and

I came also across Peres-Horodecki separability criterion, often called also PPT Criterion.

I must admit, that due to my little previous experience, I don’t understand everything well, so I’d appreciate an explanation on the abovementioned state, preferably with algebraic steps written, as they’re mostly missing in many explanations of this topic.

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My attempt

As I understand it, the very first thing is to rewrite the state-vector as a density matrix $rho$:

$$
left| psiright> = left| 0right> otimes frac{left|0 right> + ileft| 1right>}{sqrt{2}} = frac{1}{sqrt{2}} begin{bmatrix} 1 i 0 0end{bmatrix}
$$

$$
rho = left| psi right > left< psi right| = begin{bmatrix}
frac{1}{2} & -frac{i}{2} & 0 & 0
frac{i}{2} & frac{1}{2} & 0 & 0
0 & 0 & 0 & 0
0 & 0 & 0 & 0
end{bmatrix}
$$

And now I should perform "partial transform" and find eigenvalues of the partially transformed matrix. But I’m really confused here about A and B "party", as I have only one state $left| psi right>$. How am I supposed to proceed now?

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My attempt 2

I also tried to utilize a reduced density matrix:

$$
left| psi right> = frac{1}{sqrt{2}}left|00right> + frac{i}{sqrt{2}}left|01right> = frac{1}{sqrt{2}} left( left| 0right>_A otimes left| 0right>_B + ileft(left| 0right>_A otimes left| 1 right>_B right)right)
$$

$$
rho_A = Tr_B left( left|psi right> left<psi right| right) = frac{1}{2} left( left| 0right> left< 0right| + ileft( left|0right>left<1right| right) right) = frac{1}{2} begin{bmatrix}1 & i 0 & 0 end{bmatrix}
$$

$$
Tr(rho_A^2) = frac{1}{4} Trleft( begin{bmatrix} 1 & i 0 & 0 end{bmatrix} right) = frac{1}{4}
$$

As we can see, my results seems to be $frac{1}{4}$, while it should be 1 for a separable state. What am I doing wrong?