Lambert, Margrave of Tuscany
Lambert( died after 938) was the second son of Adalbert II of Tuscany and Bertha, daughter of Lothair II of Lotharingia.
He succeeded his elder brother, Guy, as count and duke of Lucca and margrave of Tuscany on his death in 928 or 929 without heirs.
In 931, before 17 October, Hugh, King of Italy, disowned and removed Lambert, giving Tuscany and the familial possession of Lucca to his brother Boso.
Hugh was Guy and Lambert's half- brother, as they had the same mother.
When Guy died, Hugh married Guy's widow, Marozia.
Abd al-Muttalib
Abd al- Muttalib Shaybah ibn Hashim( c. 497 – 578) was the grandfather of Islamic prophet Muhammad.
Teutberga
Teutberga( died 11 November 875) was a queen of Lotharingia by marriage to Lothair II.
She was a daughter of Bosonid Boso the Elder and sister of Hucbert, the lay- abbot of St. Maurice's Abbey.
Waldrada of Lotharingia
Waldrada was the mistress, and later the wife, of Lothair II of Lotharingia.
Bertha, daughter of Lothair II
Bertha( 863- 8 – March 925 in Lucca) was countess of Arles by marriage to Theobald of Arles, and margravine of Tuccany by marriage to Adalbert II of Tuscany.
She served as regent of Lucca and Tuscany from 915 until 916 during the minority of her son Guy of Tuscany.
She was described as beautiful, spirited, and courageous, and her influence over her spouse was, coupled with ambition, attributed to have involved her husbands in many wars.
She was the second illegitimate daughter of Lothair II, King of Lotharingia, by his concubine Waldrada.
Kaya Alp
Kaya Alp was, according to Ottoman tradition, the son of Kızıl Buğa and the father of Suleyman Shah, who was, in turn, the grandfather of Ertuğrul, and the great grandfather of the Ottoman Empire founder, Osman I.
Theobald of Arles
Theobald( c. 854-895), count of Arles, was a Frank from the Bosonid- family.
He was a son of Hucbert and married Bertha, an illegitimate daughter of Lothair II, King of Lotharingia, who was renowned for her beauty and spirit.
They had two sons Hugh of Italy and Boso of Tuscany.
Lothair II
Lothair II (835 –) was the king of Lotharingia from 855 until his death.
He was the second son of Emperor Lothair I and Ermengarde of Tours.
He was married to Teutberga (died 875), daughter of Boso the Elder.
Adolf I of Lotharingia
Adolf I of Lotharingia, count of Keldachgau, Vogt of Deutz from 1008 until 1018, was the son of Hermann I" Pusillus"( the Little Pfalzgraf), count palatine of Lotharingia.
He left three sons:
Lothair I
Lothair I or Lothar I (Dutch and Medieval Latin: "Lotharius", German: "Lothar", French: "Lothaire", Italian: "Lotario") (795 – 29 September 855) was the Holy Roman Emperor (817–855, co-ruling with his father until 840), and the governor of Bavaria (815–817), King of Italy (818–855) and Middle Francia (840–855).
Lothair was the eldest son of the Carolingian emperor Louis the Pious and his wife Ermengarde of Hesbaye, daughter of Ingerman the duke of Hesbaye.
On several occasions, Lothair led his full-brothers Pepin I of Aquitaine and Louis the German in revolt against their father to protest against attempts to make their half-brother Charles the Bald a co-heir to the Frankish domains.
Upon the father's death, Charles and Louis joined forces against Lothair in a three-year civil war (840–843).
The struggles between the brothers led directly to the breakup of the Frankish Empire assembled by their grandfather Charlemagne, and laid the foundation for the development of modern France and Germany.

Question: Who is Lothair Ii's paternal grandfather?
Answer: Louis the Pious