Let $\mathfrak g$ be a Lie superalgebra.

If $\mathfrak a$ is not a grade subspace of $\mathfrak g$, then why does $[\mathfrak g, \mathfrak a]$ and $[\mathfrak a, \mathfrak g]$ are not same?

For me as sets they are linear span of $[a,x]$ and $[x,a]$ and hence they are same. But in book it is given they are different and the author has defined left and right ideal separately.

I am reading the book "Lie superalgebras and enveloping algebras by Ian M.Musson" Proposition 1.2.2.

Kindly help me with this.

Thank you.