Final answer:
There is exactly one 1-dimensional Lie algebra and two distinct types of 2-dimensional Lie algebras, one of which is abelian and the other non-abelian. These algebras can all be realized as Lie subalgebras of larger Lie groups such as SO(3), SU(2), SL(2, \( \mathbb{C} \)), and GL(2, \( \mathbb{R} \)).
Step-by-step explanation:
When looking at the number of Lie algebras of a given dimension, there are straightforward classifications for low dimensions. In particular, let's consider the cases for 1-dimensional and 2-dimensional Lie algebras.
For 1-dimensional Lie algebras, there is only one, up to isomorphism, as the Lie bracket of any two elements is zero because there is only one basis element. Thus, we have a unique 1-dimensional Lie algebra, often denoted by \( \mathbb{R} \).
Moving on to 2-dimensional Lie algebras, there are two types, up to isomorphism. The first is the abelian Lie algebra, \( \mathbb{R}^2 \), where the Lie bracket of any two elements is zero. The second type is a non-abelian one, which has a Lie bracket that is non-trivial but simple enough to be defined by a single basis element acting on another. As an example, consider a basis \{ e_1, e_2 \} where [e_1, e_2] = e_1, and all other brackets vanish. This gives us the second distinct structure of a 2-dimensional Lie algebra.
All these algebras can be realized as subalgebras of larger Lie algebras. For example, any 1-dimensional subalgebra of SO(3), SU(2), SL(2, \( \mathbb{C} \)), and GL(2, \( \mathbb{R} \)) is a line through the origin in the Lie algebra, such as the span of a nonzero element in the respective basis. For the 2-dimensional algebras, one could take the plane spanned by any two linearly independent elements in the Lie algebra of these groups, making sure in the case of non-abelian to pick elements such that their bracket is non-trivial.