Victor Kac, in the paper

"Classification of infinite-dimensional simple linearly compact Lie superalgebras", http://www.mat.univie.ac.at/~esiprpr/esi605.pdf

writes at the beginning of section 5 (p.39 in the linked preprint):

A consistent $\mathbb Z$-graded Lie superalgebra of depth $\geq 2$ is infinite-dimensional (since otherwise all even elements are exponentiable).

here a consistent $\mathbb Z$-graded Lie superalgebra of depth $d$ is a complex, possibly infinite dimensional, Lie superalgebra $$ \mathfrak g=\bigoplus_{p\in\mathbb Z,\ p\geq -d} \mathfrak g_p $$ with every $\mathfrak g_p$ finite dimensional, $\mathfrak{g}_{-d}\neq 0$, and $$ \mathfrak g_{\bar 0}=\bigoplus_{p\in\mathbb Z} \mathfrak g_{2p},\quad\mathfrak g_{\bar 1}=\bigoplus_{p\in\mathbb Z} \mathfrak g_{2p+1}. $$

The superalgebra $\mathfrak g$ is also supposed to be *fundamental*, i.e. $\mathfrak g_{-1}$ generates $\oplus_{p<0}\mathfrak g_p$, and *transitive*, i.e. $[X,\mathfrak g_{-1}]=0$ implies $X=0$ for $X\in\mathfrak g_p$, $p\geq 0$.

The statement is clearly true if $\mathfrak g$ is also *even*, i.e. if all even elements that are exponentiable have nonnegative degree. The result indeed is needed only for even superalgebras, however the hypothesis is not explicitly stated. My question is the following:

*Are there finite-dimensional consistently graded fundamental transitive Lie superalgebras of depth $\geq 2$?*