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Perhaps it helps to write down all these things more explicitly:

To start, let us define $$J\equiv J_1 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2 $$ and $$J^2 \equiv J_x^2 + J_y^2 + J_z^2 \quad ,$$ where the analogous definition should hold for $J_1$ and $J_2$. Further, define for $k=x,y,z$: $$ J_k \equiv (J_1)_k \otimes \mathbb I_2 + \mathbb I_1 \otimes (J_2)_k \quad .$$

Now let us compute $J_x^2$: \begin{align} J_x^2 = J_x \, J_x &= \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \, \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \\ &= (J_1)_x \, (J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\, (J_2)_x + 2\, (J_1)_x \otimes (J_2)_x \quad . \end{align}

You can repeat the same procedure for $J_y^2$ and $J_z^2$. Adding these three terms then yields

$$J^2 = J_1^2 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2^2 + 2\, J_1 \cdot J_2 \quad , $$

where we have defined $$J_1 \cdot J_2 \equiv (J_1)_x \otimes (J_2)_x + (J_1)_y \otimes (J_2)_y+ (J_1)_z \otimes (J_2)_z \quad .$$

Perhaps it helps to write down all these things more explicitly:

To start, let us define $$J\equiv J_1 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2 $$ and $$J^2 \equiv J_x^2 + J_y^2 + J_z^2 \quad ,$$ where the analogous definition should hold for $J_1$ and $J_2$. Further, define for $k=x,y,z$: $$ J_k \equiv (J_1)_k \otimes \mathbb I_2 + \mathbb I_1 \otimes (J_2)_k$$

and let us compute \begin{align} J_k^2 \equiv J_k \, J_k &= \left((J_1)_k \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_k\right) \, \left((J_1)_k \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_k\right) \\ &= (J_1)_k \, (J_1)_k \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_k\, (J_2)_k + 2\, (J_1)_k \otimes (J_2)_k \quad . \end{align}

Adding the three contributions from $k=x,y,z$ yields

$$J^2 = J_1^2 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2^2 + 2\, J_1 \cdot J_2 \quad , $$

where we have defined $$J_1 \cdot J_2 \equiv (J_1)_x \otimes (J_2)_x + (J_1)_y \otimes (J_2)_y+ (J_1)_z \otimes (J_2)_z \quad .$$

Perhaps it helps to write down all these things more explicitly:

To start, let us define $$J\equiv J_1 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2 $$ and $$J^2 \equiv J_x^2 + J_y^2 + J_z^2 \quad ,$$ where the analogous definition should hold for $J_1$ and $J_2$. Now let us compute $J_x^2$: \begin{align} J_x^2 = J_x \, J_x &= \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \, \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \\ &= (J_1)_x \, (J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\, (J_2)_x + 2\, (J_1)_x \otimes (J_2)_x \quad . \end{align}

You can repeat the same procedure for $J_y^2$ and $J_z^2$. Adding these three terms then yields

$$J^2 = J_1^2 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2^2 + 2\, J_1 \cdot J_2 \quad , $$

where we have defined $$J_1 \cdot J_2 \equiv (J_1)_x \otimes (J_2)_x + (J_1)_y \otimes (J_2)_y+ (J_1)_z \otimes (J_2)_z \quad .$$

Perhaps it helps to write down all these things more explicitly:

To start, let us define $$J\equiv J_1 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2 $$ and $$J^2 \equiv J_x^2 + J_y^2 + J_z^2 \quad ,$$ where the analogous definition should hold for $J_1$ and $J_2$. Further, define for $k=x,y,z$: $$ J_k \equiv (J_1)_k \otimes \mathbb I_2 + \mathbb I_1 \otimes (J_2)_k \quad .$$

Now let us compute $J_x^2$: \begin{align} J_x^2 = J_x \, J_x &= \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \, \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \\ &= (J_1)_x \, (J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\, (J_2)_x + 2\, (J_1)_x \otimes (J_2)_x \quad . \end{align}

You can repeat the same procedure for $J_y^2$ and $J_z^2$. Adding these three terms then yields

$$J^2 = J_1^2 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2^2 + 2\, J_1 \cdot J_2 \quad , $$

where we have defined $$J_1 \cdot J_2 \equiv (J_1)_x \otimes (J_2)_x + (J_1)_y \otimes (J_2)_y+ (J_1)_z \otimes (J_2)_z \quad .$$

Perhaps it helps to write down all these things more explicitly:

To start, let us define $$J\equiv J_1 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2 $$ and $$J^2 \equiv J_x^2 + J_y^2 + J_z^2 \quad ,$$ where the analogous definition should hold for $J_1$ and $J_2$. Now let us compute $J_x^2$: \begin{align} J_x^2 = J_x \, J_x &= \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \, \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \\ &= (J_1)_x \, (J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\, (J_2)_x + 2\, (J_1)_x \otimes (J_2)_x \quad . \end{align}

You can repeat the same procedure for $J_y^2$ and $J_z^2$. Then adding all three terms yields

$$J^2 = J_1^2 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2^2 + 2\, J_1 \cdot J_2 \quad , $$

where we have defined $$J_1 \cdot J_2 = (J_1)_x \otimes (J_2)_x + (J_1)_y \otimes (J_2)_y+ (J_1)_z \otimes (J_2)_z \quad .$$

Perhaps it helps to write down all these things more explicitly:

To start, let us define $$J\equiv J_1 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2 $$ and $$J^2 \equiv J_x^2 + J_y^2 + J_z^2 \quad ,$$ where the analogous definition should hold for $J_1$ and $J_2$. Now let us compute $J_x^2$: \begin{align} J_x^2 = J_x \, J_x &= \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \, \left((J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\right) \\ &= (J_1)_x \, (J_1)_x \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes (J_2)_x\, (J_2)_x + 2\, (J_1)_x \otimes (J_2)_x \quad . \end{align}

You can repeat the same procedure for $J_y^2$ and $J_z^2$. Adding these three terms then yields

$$J^2 = J_1^2 \otimes \mathbb{I}_2 + \mathbb{I}_1 \otimes J_2^2 + 2\, J_1 \cdot J_2 \quad , $$

where we have defined $$J_1 \cdot J_2 \equiv (J_1)_x \otimes (J_2)_x + (J_1)_y \otimes (J_2)_y+ (J_1)_z \otimes (J_2)_z \quad .$$