The rank-1 Racah algebra $R\left(3\right)$ plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the 3-parameter system on the 2-sphere from which all second-order conformally flat superintegrable models in 2D can be obtained by means of suitable limits and contractions. A higher rank generalisation of $R\left(3\right)$, the so-called rank $n-2$ Racah algebra $R\left(n\right)$, has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the $(n-1)$-sphere. In the present work, we show that such an algebraic structure naturally arises as embedded inside a larger quadratic algebra characterising $n$D superintegrable models with non-central terms. This is shown both in classical and quantum mechanics through suitable (symplectic or differential) realisations of the Racah and additional generators. Among the main results, we present an explicit construction of the complete symmetry algebras for two families of $n$-dimensional maximally superintegrable models, the Smorodinsky–Winternitz system and the generalised Kepler–Coulomb system. For both families, the underlying symmetry algebras are higher-rank quadratic algebras containing the Racah algebra $R\left(n\right)$ as subalgebra. High-order algebraic relations among the generators of the full quadratic algebras are also obtained both in the classical and quantum frameworks. These results should shed new light to the further understanding of the structures of quadratic algebras in the context of superintegrable systems.

等级1的Racah代数 $\mathrm{[R}（3）$在超积分系统的理论中起着举足轻重的作用。它似乎是2球面上3参数系统的对称代数，可以通过适当的限制和收缩从中获得所有二维2D保形平坦的超积分模型。更高级别的概括$\mathrm{[R}（3）$，所谓的等级 $ñ-2$ 拉卡代数 $\mathrm{[R}（ñ）$，最近被考虑，并且证明是上的一般超可积模型的对称代数。 $（ñ-\mathrm{1个}）$-领域。在目前的工作中，我们证明了这样的代数结构自然会嵌入到较大的二次代数中$ñ$具有非中心项的D超可积模型。古典力学和量子力学都通过Racah和其他生成器的适当实现（折衷或微分）实现了这一点。在主要结果中，我们提出了两个族的完全对称代数的显式构造$ñ$维最大超积分模型，Smorodinsky-Winternitz系统和广义开普勒-Coulomb系统。对于两个族，基本对称代数都是包含Racah代数的高阶二次代数$\mathrm{[R}（ñ）$作为子代数。在经典框架和量子框架中，还获得了完整二次代数生成器之间的高阶代数关系。这些结果将为进一步理解超积分系统中的二次代数结构提供新的思路。