inhomogeneous elements in vector superspace?

Question

I am currently studying Lie Superalgebra, and having a confusion on basic concepts of superalgebra. From Kac's Lie Superalgebra(p.13), it says that "if deg$a$ appears in a given Superalgebra, then it is assumed that $a$ is homogeneous, and that the expression is extended to the other elements by the linearity". What I am confused is that in any superalgebra, is there a inhomogeneous elements in Superalgebra i.e which is neither even nor odd? Thanks

Answer 1

Almost any element of a superalgebra $\mathfrak g$ is inhomogeneous. In fact, if $\mathfrak g_0$ is the even part of $\mathfrak g$ and if $\mathfrak g_1$ is the odd part, then for every $X\in\mathfrak g_0\setminus\{0\}$ and for every $Y\in\mathfrak g_1\setminus\{0\}$, $X+Y$ is inhomogeneous.