In the form, $$[J_i,J_j]=i\epsilon_{ijk}J_k\tag{1}$$ the Lie algebra of ${\rm SO(3)}$, denoted by $\mathfrak{so}(3)$, is called real Lie algebra.

By taking complex linear combinations $J_{\pm}=J_1\pm iJ_2$, $(1)$ can be written in the form $$[J_3,J_{\pm}]=\pm 2J_{\pm},~~~ [J_+,J_-]=2J_3.\tag{2}$$ Now, it is called the complexified Lie algebra of ${\rm SO(3)}$, denoted by $\mathfrak{so}(3)_{\mathbb{C}}$.

Question $1$ In what sense the algebra $(1)$ is real but $(2)$ is complex(ified)? Essentially, I am asking, what was so real about $(1)$ that has become complex in $(2)$?

Question $2$ From the point of view of representation theory (as applied to physics), why is it necessary to differentiate real and complexified Lie algebras?

I did look at a couple of similar posts, in particular,

"How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$?" and,

"Motivating Complexification of Lie Algebras?".

But I think, here I am asking a more elementary question than these posts seem to deal with.

In the form, $$[J_i,J_j]=i\epsilon_{ijk}J_k\tag{1}$$ the Lie algebra of ${\rm SO(3)}$, denoted by $\mathfrak{so}(3)$, is called real Lie algebra.

By taking complex linear combinations $J_{\pm}=J_1\pm iJ_2$, $(1)$ can be written in the form $$[J_3,J_{\pm}]=\pm 2J_{\pm},~~~ [J_+,J_-]=2J_3.\tag{2}$$ Now, it is called the complexified Lie algebra of ${\rm SO(3)}$, denoted by $\mathfrak{so}(3)_{\mathbb{C}}$.

Question $1$ In what sense the algebra $(1)$ is real but $(2)$ is complex(ified)? Essentially, I am asking, what was so real about $(1)$ that has become complex in $(2)$?

Addendum The issue is, given a Lie algebra structure [such as $(1)$ or $(2)$], how does one figure out whether it is a real Lie algebra of the group or a complexified one?

Question $2$ From the point of view of representation theory (as applied to physics), why is it necessary to differentiate real and complexified Lie algebras?

I did look at a couple of similar posts, in particular,

"How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$?" and,

"Motivating Complexification of Lie Algebras?".

But I think, here I am asking a more elementary question than these posts seem to deal with.

In the form, $$[J_i,J_j]=i\epsilon_{ijk}J_k\tag{1}$$ the Lie algebra of ${\rm SU(2)}$ or ${\rm SO(3)}$, denoted by $\mathfrak{su}(2)$ or $\mathfrak{so}(3)$, is called real Lie algebra. However, as far as I understand, written in the form $$[J_z,J_{\pm}]=\pm 2J_{\pm},~~~ [J_+,J_-]=2J_z,\tag{2}$$ it is called the complexified Lie algebra, denoted by $\mathfrak{su}(2)_{\mathbb{C}}$ or $\mathfrak{so}(3)_{\mathbb{C}}$. Please correct me if I am wrong.

Question $1$ In what sense the algebra $(1)$ is real but $(2)$ is complex(ified)? Essentially, I am asking, what was so real about $(1)$ that has become complex in $(2)$?

Question $2$ From the point of view of representation theory (as applied to physics), why is it necessary to differentiate real and complexified Lie algebras?

I did look at a couple of similar posts, in particular,

"How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$?" and,

"Motivating Complexification of Lie Algebras?".

But I think, here I am asking a more elementary question than these posts seem to deal with.

In the form, $$[J_i,J_j]=i\epsilon_{ijk}J_k\tag{1}$$ the Lie algebra of ${\rm SO(3)}$, denoted by $\mathfrak{so}(3)$, is called real Lie algebra.

By taking complex linear combinations $J_{\pm}=J_1\pm iJ_2$, $(1)$ can be written in the form $$[J_3,J_{\pm}]=\pm 2J_{\pm},~~~ [J_+,J_-]=2J_3.\tag{2}$$ Now, it is called the complexified Lie algebra of ${\rm SO(3)}$, denoted by $\mathfrak{so}(3)_{\mathbb{C}}$.

Question $1$ In what sense the algebra $(1)$ is real but $(2)$ is complex(ified)? Essentially, I am asking, what was so real about $(1)$ that has become complex in $(2)$?

Question $2$ From the point of view of representation theory (as applied to physics), why is it necessary to differentiate real and complexified Lie algebras?

I did look at a couple of similar posts, in particular,

"How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$?" and,

"Motivating Complexification of Lie Algebras?".

But I think, here I am asking a more elementary question than these posts seem to deal with.

In the form, $$[J_i,J_j]=i\epsilon_{ijk}J_k\tag{1}$$ the Lie algebra of ${\rm SU(2)}$ or ${\rm SO(3)}$, denoted by $\mathfrak{su}(2)$ or $\mathfrak{so}(3)$, is called real Lie algebra. However, as far as I understand, written in the form $$[J_z,J_{\pm}]=\pm 2J_{\pm},~~~ [J_+,J_-]=2J_z,\tag{2}$$ it is called the complexified Lie algebra, denoted by $\mathfrak{su}(2)_{\mathbb{C}}$ or $\mathfrak{so}(3)_{\mathbb{C}}$. Please correct me if I am wrong.

Question $1$ In what sense the algebra $(1)$ is real but $(2)$ is complex(ified)? Essentially, I am asking, what was so real about $(1)$ that has become complex in $(2)$?

Question $2$ From the point of view of representation theory (as applied to physics), why is it necessary to differentiate real and complexified Lie algebras?

I did look at a couple of similar posts, in particular,

"How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$?" and,

"Motivating Complexification of Lie Algebras?".

But I think, here I am asking a more elementary question than these posts seem to deal with.

In the form, $$[J_i,J_j]=i\epsilon_{ijk}J_k\tag{1}$$ the Lie algebra of ${\rm SU(2)}$ or ${\rm SO(3)}$, denoted by $\mathfrak{su}(2)$ or $\mathfrak{so}(3)$, is called real Lie algebra. However, as far as I understand, written in the form $$[J_z,J_{\pm}]=\pm 2J_{\pm},~~~ [J_+,J_-]=2J_z,\tag{2}$$ it is called the complexified Lie algebra, denoted by $\mathfrak{su}(2)_{\mathbb{C}}$ or $\mathfrak{so}(3)_{\mathbb{C}}$. Please correct me if I am wrong.

Question $1$ In what sense the algebra $(1)$ is real but $(2)$ is complex(ified)? Essentially, I am asking, what was so real about $(1)$ that has become complex in $(2)$?

Question $2$ From the point of view of representation theory (as applied to physics), why is it necessary to differentiate real and complexified Lie algebras?

I did look at a couple of similar posts, in particular,

"How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$?" and,

"Motivating Complexification of Lie Algebras?".

But I think, here I am asking a more elementary question than these posts seem to deal with.