This paper is aimed at determining the derivation superalgebra of modular Lie superalgebra $\overline{K}(n,m)$. To that end, we first describe the $\mathbb{Z}$-homogeneous derivations of $\overline{K}(n,m)$. Then we obtain the derivation superalgebra $Der(\overline{K})$. Finally, we partly determine the derivation superalgebra $Der(K)$ by virtue of the invariance of $K(n,m)$ under $Der(\overline{K})$.

In this paper, we mainly study super-bimodules on Bihom-Jordan superalgebras and present some interesting constructions from the perspective of super-bimodules. Meanwhile, we give abelian extension through super-bimodules. In addition, we give the representations, $\mathcal{O}$-operators and Rota–Baxter operators of Bihom-Jordan superalgebras. Specially, using $\mathcal{O}$-operators, we characterize Bihom-pre-Jordan superalgebras.

This paper presents a classification of real matrix representations of Rota-Baxter operators on the real Lie algebra $\mathfrak{s}\mathfrak{o}(3)$. We obtained all non-isomorphic classical Rota-Baxter operators of weight $0$ up to orthogonal similarity, as well as special Rota-Baxter operators (multiplicative and pseudo-Rota-Baxter operators) of weights $0$ and $1$. For weight $0$, we found three canonical symmetric matrix forms that solve the classical Yang-Baxter equation and used them to construct explicit left-symmetric algebra structures. Interestingly, no non-trivial multiplicative Rota–Baxter operators exist on $\mathfrak{s}\mathfrak{o}(3)$ for weights $0$ or $1$. Pseudo-Rota-Baxter operators exhibit a rich structure: for weight 1, we found two families of symmetric matrices, while for weight 0, there were five families parameterized by continuous parameters $g_{11}\in [0,1]$ and $g_{22}\in [\frac{1}{2},1]$. These findings contribute to the theoretical exploration of Rota-Baxter operators on real orthogonal Lie algebras by presenting explicit $3\times 3$ matrix solutions, which offer valuable tools for their practical implementation in robotic mechanisms.