Understanding the Symmetry of Field Strength Tensors in Metric Properties

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\partialμgλρ - \partialρgμλ = 0

I have reaced to this statement. Is this true? How?

What I thought is it looks like field strength tensor, but symmetric hence we can say it is true.

Thank you.
 
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felixphysics said:
\partialμgλρ - \partialρgμλ = 0

I have reaced to this statement. Is this true? How?

It's not generally true.
 

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