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Definition & Example

An irreducible finite Cartan type is one of the following:
 $A_n$ for $n \geq 1$;
 $B_n$ for $n \geq 2$;
 $C_n$ for $n \geq 3$;
 $D_n$ for $n \geq 4$;
 $E_n$ for $n = 6,7,8$;
 $F_4$;
 $G_2$;

The rank of an irreducible finite Cartan type is its index. It denotes the dimension of the corresponding irreducible representation of its Weyl group.
the 5 Finite Cartan types of size 8  
['A',8]  
['B',8]  
['C',8]  
['D',8]  
['E',8] 
Additional information
Finite root systems
Let $V$ be an Euclidean vector space of finite dimension, endowed with an inner product $(\cdot, \cdot)$, and let $\alpha\in V$ with $\alpha \neq 0$. The reflection $s_\alpha:V\rightarrow V$ orthogonal to $\alpha$ is defined as
 $\Phi \cap \mathbb{R}\alpha = \{\alpha, \alpha\}$,
 $s_\alpha \Phi = \Phi$ for all $\alpha \in \Phi$ and
 $\langle\beta,\alpha\rangle := 2\frac{(\beta,\alpha)}{(\alpha,\alpha)} \in \mathbb{Z}$ for all $\alpha,\beta\in \Phi$.
The elements of $\Phi$ are called roots. The root system $\Phi$ is called reducible, if there is a disjoint union $\Phi = \Phi_1 \sqcup \Phi_2$ in nonempty sets such that $(\alpha,\beta) = 0$ for all $\alpha \in \Phi_1$ and $\beta \in \Phi_2$. Otherwise it is called irreducible.
Associated to a root system is its Weyl group
Let $\Phi$ be a root system. A subset $\Delta \subset \Phi$ is called a simple system if it is a basis of $V$ and every root in $\Phi$ is a linear combination of elements of $\Delta$ with integral coefficients all of the same sign (or zero).
Associated to a simple system $\Delta$ is the positive system $\Phi^+$ containing all roots that are linear combinations of $\Delta$ with only nonnegative coefficients.
Elements of $\Delta$ are called simple roots and elements of $\Phi^+$ are called positive roots. For every root system, simple systems exist, see [Hum72].
Let $\Delta = \{\alpha_1,\dots,\alpha_n\}$ be a simple system of $\Phi$. The Dynkin diagram of $\Phi$ is defined as a graph on $n$ vertices where each pair $(i,j)$ is connected by $\langle\alpha,\beta\rangle\langle\beta,\alpha\rangle$ edges. Whenever $i$ and $j$ are connected by more than one edge, add an arrow indicating which root is of greater length. The Dynkin diagram only depends on $\Phi$ and not on the choice of $\Delta$, see [Hum72].
Irreducible crystallographic root systems are classified by Dynkin diagrams and named by finite Cartan types, see here.
Noncrystallographic finite root systems
Dropping condition (3) in the definition extends the class of root systems by the noncrystallographic cases. This gives a classification of finite Coxeter groups, see [Hum92]. In addition to the classification of the cystallographic irreducible root systems (and their associated Weyl groups), there are finite Coxeter groups of types $I_2(m)$ (symmetry group of a regular $m$gon) and types $H_3$ (symmetry group of the regular icosahedron and dodecahedron) and $H_4$ (symmetry group of the 120cell and of the 600cell).
Other objects classified by finite Cartan types
Semisimple Lie Algebras
Let $F$ be an algebraically closed field of characteristic zero. A Lie algebra is a vector space $\mathfrak{g}$ over $F$ together with an operation $\mathfrak{g}\times \mathfrak{g}\rightarrow \mathfrak{g}, (x,y) \mapsto [x,y]$, called Lie bracket, such that the following is satisfied:
 The bracket operation is bilinear;
 $[x,x] = 0$ for all $x\in \mathfrak{g}$;
 $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$ for all $x,y,z\in \mathfrak{g}$.
A subspace $I$ of $\mathfrak{g}$ is called an ideal of $\mathfrak{g}$ if $[x,y]\in I$ for all $x\in \mathfrak{g}, y\in I$. A special ideal of $\mathfrak{g}$ is the derived algebra $[\mathfrak{g},\mathfrak{g}]$ consisting of all possible brackets of elements from $\mathfrak{g}$. A Lie algebra is called simple if $[\mathfrak{g},\mathfrak{g}]\neq 0$ and $0$ and $\mathfrak{g}$ are its only ideals.
A Lie algebra $\mathfrak{g}$ is called abelian if the bracket vanishes on $\mathfrak{g}$, i.e. $[\mathfrak{g},\mathfrak{g}]=0$. It is called solvable if the derived series
If $\mathfrak{g}$ is of finite dimension, then it has a unique maximal solvable ideal, called its radical $\operatorname{Rad} \mathfrak{g}$.
A semisimple Lie algebra is a Lie algebra $\mathfrak{g}$ such that $\operatorname{Rad} \mathfrak{g} = 0$. This is equivalent to a decomposition
Given a semisimple Lie algebra L, there is a canonical way to construct a root system $\Phi$ associated to $\mathfrak{g}$ that completely determines the structure of $\mathfrak{g}$. Furthermore, a simple Lie algebras is associated to an irreducible root system. Thus, semisimple Lie algebras are completely classified by the finite Cartan types. Details on this can be found in [Hum72].
Quiver Representations
A quiver a directed graph $Q=(V,E)$ with possibly multiple edges and loops. For a ring $R$, a representation of $Q$ over $R$ is an assignment of a $R$module $R_v$ to each vertex $v\in V$ and a linear map $f_{v,w}:R_v\rightarrow R_w$ to each edge $(v,w) \in E$. A representation of $Q$ is indecomposable if it is not a sum of smaller nontrivial representations of $Q$.
Let $F$ be an algebraically closed field. Gabriel's Theorem states that a quiver $Q$ has only finitely many nonisomorphic representations of finite dimension if and only if the underlying undirected graph $\bar Q$ is of type $A$, $D$ or $E$. For a thorough introduction into quiver representations see [ASS06]
Cluster algebras of finite type
Cluster algebras were introduced by Fomin and Zelevinsky in the early 2000s in [FZ02] and they obtained the classification of cluster algebras of finite type in [FZ03].
References
 [ASS06] Ibrahim Assem, Daniel Simson and Andrzej SkowroĊski, Elements of the Representation Theory of Associative Algebras Vol. 1, Cambridge University Press (2006)
 [FZ02] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, Journal of the American Mathematical Society, 15(2), 497529 (2002)
 [FZ03] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Inventiones Mathematicae 154(1), 63121 (2003)
 [Hum72] James E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer (1972)
 [Hum92] James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press (1992)
Sage examples
Technical information for database usage
 The database contains all irreducible finite Cartan types up to rank $8$.