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[Community Question] Linear-algebra: partial trace of a subsystem

One of our user asked:

Let there exists a pure composite system of 2 subsystems; namely, "1" and "2". Suppose $\hat{\rho} = \frac{|00\rangle \langle 00|+|00\rangle\langle 11| + |11\rangle\langle 00| + |11\rangle \langle 11|}{2}$ is a pure state operator for the this pure composite system.

The reduce state operator for subsystem "1" is

$\hat{\rho}^{1} = Tr_{2}\left(\hat{\rho}\right)$

= $\frac{1}{2} Tr_{2} \left(|0\rangle\langle 0|\otimes|0\rangle \langle 0| + |1\rangle\langle0|+|0\rangle\langle1| \otimes|0\rangle\langle 1| \otimes |0\rangle\langle1|+|1\rangle\langle1|\otimes|1\rangle\langle1| \right)$

= $\frac{1}{2}\left(|0\rangle \langle0|\cdot Tr\left(|0\rangle \langle 0|\right) + \cdot \cdot \cdot \right)$

I am unable to understand how the third inequality comes about. Any help is appreciated.


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