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.



left ideals in Lie super algebras
The linear spans of $[a,x]$ and $[x,a]$, in a Lie superalgebra (i.e. a $\mathbb{Z}\_2$-graded Lie algebra) are generally not the same (unlike the Lie algebras case):


Since $\mathfrak a$ is not a graded subspace of $\mathfrak g$, then in general its elements are not homogeneous. So for $a\in \mathfrak a$ we generally have: $a=a\_0+a\_1$, i.e. its decomposition into even and odd parts.  

Thus, if we take for example an odd element $x\in \mathfrak g\_1$:
$$
[a,x]=[a\_0,x]+[a\_1,x]=-[x,a\_0]+[x,a\_1]
$$
while
$$
[x,a]=[x,a\_0]+[x,a\_1]
$$
Consequently, the elements $[a,x]$ and $[x,a]$, are -in general- not collinear, as has already been indicated in the comments to the OP. 


