  In this paper, we give some conditions, under which, if an infinitely
divisible distribution supported on $[0,\infty)$ belongs to the intersection of
exponential distribution class $\mathcal{L}(\gamma)$ for some $\gamma\ge0$ and
generalised subexponential distribution class $\mathcal{OS}$, then its
L$\rm\acute{e}$vy spectral distribution or convolution of the distribution with
itself also belongs to the same one. To this end, we discuss the closure under
the compound convolution roots for the class. In addition, we do some in-depth
discussion about the above-mentioned conditions, and provide some types of
distributions satisfying them. Further, we obtain some local versions of the
above-mentioned results by the Esscher transform of distributions. Therefore,
some positive conclusions related to the Embrechts-Goldie conjecture are
obtained. Prior to this, all corresponding results are negative
