# Definition A *Lie algebra isomorphism* is a [[Lie Algebra Homomorphism]] that is also a [[Vector Space Isomorphism]]. If two [[Lie Algebra|Lie algebras]] are [[isomorphism|isomorphic]], then they are equivalent and we write $\mathfrak{g} \cong \mathfrak{h}$. Note that being isomorphic implies that the Lie algebras have the same [[dimension]]. # Structure Constants If we pick a [[basis]] $\{X_i\}_{i=1\ldots n}$ for $\mathfrak{g}$ and $\{Y_i\}_{i=1\ldots n}$ for $\mathfrak{h}$, then the commutation relations take the form: $ \begin{align} [X_i, X_j] &= \sum_{k=1}^n c_{ij}^{\,\,\,\,k}X_k\\ [Y_i, Y_j] &= \sum_{k=1}^n d_{ij}^{\,\,\,\,k}Y_k \end{align} $ If the [[Structure Constants of a Lie Algebra|structure constants]] are equal $c_{ij}^{\,\,\,\,k} = d_{ij}^{\,\,\,\,k}, \, \forall\, i,j,k$ then we can find a map: $ \begin{align} \phi: \mathfrak{g}&\rightarrow\mathfrak{h}\\ v^iX_i&\mapsto v^iY_i \end{align} $ which is a Lie algebra isomorphism.