Let \(C \ell _{d}^{\frac{1}{3}}\) be the complex ternary Clifford algebra on d generators as defined in Cerejeiras and Vajiac (Adv Appl Clifford Algebras 31:13, 2021). The main objective of this work is to investigate algebraic structure of these algebras and present a new Structure Theorem. In particular, it is shown that when d is even, the algebras are central simple (CSA/\({\mathbb {C}}\)), and when d is odd \((d>1)\) then the algebras are direct sums of three simple ideals, each being isomorphic to a ternary Clifford algebra \(C \ell _{d-1}^{\frac{1}{3}}\). In the latter case, each simple ideal is generated by a central primitive idempotent as, for each d, there are exactly three central primitive orthogonal idempotents which decompose the algebra unit. A formula is given for computing these idempotents for each odd d \((d>1)\). We conclude that \(C \ell _{2k}^{\frac{1}{3}} \cong \mathop {\mathrm{Mat}}\nolimits (3^k,{\mathbb {C}})\) and \(C \ell _{2k+1}^{\frac{1}{3}} \cong {}^3 \mathop {\mathrm{Mat}}\nolimits (3^k,{\mathbb {C}})\). We describe an algorithm for finding a complete set of \(3^k\) orthogonal primitive idempotents in \(C \ell _{2k}^{\frac{1}{3}}\). Each such idempotent generates a minimal left (or right) ideal which carries an irreducible faithful representation of \(C \ell _{2k}^{\frac{1}{3}}\). This allows us then to find irreducible representations of \(C \ell _{2k+1}^{\frac{1}{3}}\) in minimal left (or right) ideals. This paper is a continuation of Abl amowicz (Adv Appl Clifford Algebras 31: 62, 2021).

令\(C \ell _{d}^{\frac{1}{3}}\)为 Cerejeiras 和 Vajiac 中定义的d生成元上的复三元 Clifford 代数（Adv Appl Clifford Algebras 31:13, 2021）。这项工作的主要目的是研究这些代数的代数结构并提出一个新的结构定理。特别是，当d为偶数时，代数是中心简单的 (CSA/ \({\mathbb {C}}\) )，当d为奇数时\((d>1)\)，则代数是三个简单理想的直接和，每个理想都同构于三元 Clifford 代数\(C \ell _{d-1}^{\frac{1}{3}}\). 在后一种情况下，每个简单理想都是由一个中心原语幂等生成的，因为对于每个d，恰好有三个中心原语正交幂等分解代数单元。给出了一个公式来计算每个奇数d \((d>1)\)的这些幂等。我们得出结论\(C \ell _{2k}^{\frac{1}{3}} \cong \mathop {\mathrm{Mat}}\nolimits (3^k,{\mathbb {C}})\ )和\(C \ell _{2k+1}^{\frac{1}{3}} \cong {}^3 \mathop {\mathrm{Mat}}\nolimits (3^k,{\mathbb { C}})\)。我们描述了一种在\(C \ell _{2k}^{\frac{1}{3}}\)中找到一组完整的\(3^k\)正交基元幂等的算法. 每个这样的幂等生成一个最小的左（或右）理想，它带有\(C \ell _{2k}^{\frac{1}{3}}\)的不可约的忠实表示。这使我们能够在最小左（或右）理想中找到\(C \ell _{2k+1}^{\frac{1}{3}}\)的不可约表示。本文是 Abl amowicz (Adv Appl Clifford Algebras 31: 62, 2021) 的续篇。